 Okay, so so our final talk of this session is going to be given by Laura Ratcliffe from Bristol Talking about simulating organic LEDs With large-scale DFT. So floor is yours Laura Okay, thank you very much and thanks to the organizers for inviting me here. It's great to be back in Tres I haven't been here since my PhD. So it's been a while Yep, so I'm going to be talking about simulating disorder in organic LEDs using large-scale DFT So just a brief overview. So I'm going to start with a motivation of why OLEDs why we're interested in disorder Why we need large-scale DFT for that and I'm then going to talk about how we do large-scale DFT So by large here I mean that the thousands of atoms regime and our particular flavor of doing that using a wavelet based a set approach And then I'm going to focus a bit more time on how we go beyond the ground state Which is important for the type of OLEDs that I'm interested in and then very briefly touch on an example of a particular type of OLED So the motivation so I mean this doesn't really need explaining to this audience I'm sure we're all familiar with the concept of trying to do computational materials discovery The idea being that we want to somehow narrow down the search space for experimentalists who are looking for New materials which might work better for a particular application Whether that's kind of cheaper or more environmentally friendly or more efficient. It depends on the application in question And so I'm particularly interested in organic electronics So these have a number of advantages being that they're lightweight low cost They can also be flexible so that leads to some interesting possibilities for devices Whether that's things like the curve TV here or the idea of kind of foldable and and bendy screens But one of the the challenges for organic electronics is so we we need to do kind of want to do this this workflow Of course, we need to balance kind of two things one that the cost of our calculations If we want to screen a very large number of potential molecules Then we need our calculations to be cheap enough to do that But we also need to balance that with the accuracy of course if our our calculations are not accurate enough We're not going to be able to focus the search space in the way that we want to do so So the particular challenge for OLED so one of the properties that people are often interested in for charge Transport is things like on-site energies and transfer integrals and the usual approach of doing this is to start with a Typical OLED structure, which is often very very disordered like this kind of snapshot you see here And typically we might extract pairs of molecules and calculate the transfer integrals for these pairs And one kind of limitation of this is that we are taking into account this disorder But we're not including the effect of the environment on these parameters And in some cases the environment can have quite an important effect So if we want to push more in the direction of accuracy, then we need to be able to do large systems But at the same time we need to be aware of this need to balance the computational cost And so I'm interested in a particular class of OLEDs Which have kind of been relatively recently discovered based on a process called thermally activated delayed fluorescence or TADF So the motivation for TADF is that most kind of OLED emitters in the current generation Or you can't see it because I've crossed it out but are based on heavy metals like iridium or platinum And this is not good from an environmental perspective So we'd like to move towards something which could be purely organic The advantage of TADF is that it can be purely organic and it's basically Based on this process of reversing the system crossing between the triplet and the singlet level And if this singlet triplet splitting is small enough and this can be firmly activated So we essentially get this delayed fluorescence process as well as the phosphorescence So it's quite an efficient process and as I said, it can be purely organic So if we want to be able to find new TADF emitters our key kind of parameter that we're interested in is this singlet triplet splitting But so the challenge here particularly is that the singlet triplet splitting Both the singlet and the triplet can be some combination of charge transfer and local excitations So we need to really be able to handle both classes of excitations And as with the charge transport parameters, the environment can also have a particular effect on the singlet triplet splitting So really we'd like to be able to look at excitations in large systems So this is an example of a TADF emitter to CZPN Again, you can see a disordered structure and we also have this Internal molecular disorder that we have this rotation around these two angles And we have all sorts of other distortions going on So there's an opportunity to look at the ground state properties But also particularly the excited state properties if we want to understand how the disorder and environment are important for TADF And so that's why I'm interested in large systems. So the question is how and so I work with a code called big DFT Which is based on the wavelet basis set Which is a bit unusual in the DFT field and in big DFT particular we use this type of Wavelet basis set which are called the least asymmetric Dilbysheys of order 16 It's a bit of a mouthful, but basically they're these two squiggly functions The red one is the scaling function the blue one is the wavelet and these have some nice properties in that they're orthogonal They're localized in both real and furry space and then more particularly we can use them to define a multi resolution approach So in big DFT that means we have this regular grid We have some what we call fine grid points closer to the atoms where the wave function is strongly varying And here we have both scaling functions and wavelets and then further away We have what we call the coarse grid where the wave function is less strongly varying and we just have scaling functions So this leads to quite an efficient approach particularly if you have inhomogeneous systems And we can also work with a range of boundary conditions So for molecules through to fully periodic solids in this particular work for the OLEDs I'm mostly working with with clusters in free boundary conditions And so it's an open source code in case you're interested in using it But despite the nice properties of wavelets it still is a cubic scaling code So just like with plane waves the computational and cost increases as than a cube of the number of atoms in the system So this is still kind of a limitation that Typically we can go up to maybe a thousand atoms maybe a few more And we're mostly limited actually by that the cubic scaling cost of the memory, but we'd really like to be able to go further And so how do we do that? So this is just a one-slide introduction to linear scaling big DFT And so the idea of linear scaling approach is that they kind of want to exploit the principle of near sightedness So the idea behind near sightedness is we're basically saying that the atom door electrons Don't really care about what's going on far away from them So we should be able to somehow introduce some localization in our system And in big DFT we introduced localize what we call support functions So this is approach that's used in other linear scaling codes like one type of conquest for example And these are basically numerical functions which are represented in wavelets So this is an example of what they look like for this particular molecule So we have s-like functions and p-like functions, you know, so potentially d-like functions But you can see that they're not strictly s orbitals. They are fully numerical And so they don't kind of exactly retain the symmetry of s and p orbitals And so once we have these support functions We can write out our density matrix in terms of the support functions and what we call the density kernel So these are our two main quantities that we use in linear scaling big DFT And then the algorithm goes basically we start with some atomic orbital guess We optimize our support function So this means we minimize the energy of the system with respect to the numerical form of the support functions So this basically enables us to start with some to have some small basis. So For this system, we would literally have four basis functions on the carbon atom so 1s and 3p orbitals and then we optimize them to adapt to the system So they essentially adapt some way to reflect their local chemical environment And if you look closely you can kind of see this is why we get slightly different s-like orbitals on carbon atoms with different environments And so these changes are actually quite subtle So this 1d projection here the dash lines are showing our linear combination of atomic orbitals guess the straight lines show the Optimized orbitals so the changes are quite subtle, but this is quite important for the accuracy and so we do end up with a minimal basis which is Accurate and has an accuracy kind of similar to that of the underlying wavelets And we also have a tunable parameter that if you increase the radius of these support functions You get closer and closer to the cubic scaling result And so this enables us already to go to quite big systems But if we're going to be doing lots of calculations We want to reduce the cost as much as we possibly can and so this led us to kind of think about what's the bottleneck? So since I showed so we're optimizing the support functions and the kernel But the support function optimization really takes quite a lot of our compute computational time But we know that these support functions adapt to their local chemical environment So if we would look at for example here a nano tube You can see that along the tube in the same environment the support functions look identical Or in a kind of molecular system like this little water droplet again The support functions on one water molecule look very similar to those on another water molecule So we have the idea that we should be able to somehow reuse this to reduce the cost of our calculations So this led us to define what we call a molecular fragment approach So the idea here is that we start with some representative fragment or molecule in our system We optimize the support functions for that one molecule Which is very cheap because it's a small calculation And then we replicate them and use them as a fixed basis for our larger system So here it's a water droplet in the applications. I'm interested in it's going to be a cluster of OLED molecules So this enables us to get rid of this loop which is going to lead to some quite big savings in computational cost And there's just one more ingredient that we need so this these functions as I mentioned the numerical They're based on a grid so we need first of all to understand how we can do this rotor translation And it turns out there's a well-established way of doing this We just minimize this cost function which basically gives us rotation matrix between some templates or initial coordinates and some final system coordinates and The nice thing about this cost function is that if it's a rigid transformation Then the cost function will be zero if you have some distortions in the system So if your water molecules are slightly fluctuating this will be non-zero So it also gives you some idea of the kind of distortions in your system And then we just do a wavelet based interpolation to actually get from one particular orientation to another And so when does this work? So there's two kind of caveat So first of all we need our fragments to be relatively weakly interacting If you would have fragments which are very close together or linked by a covalent bond Then it doesn't make sense to use a basis set which was generated for isolated fragments And then secondly we also want the distortions to not be too big So if you would have some molecule and you significantly change the bond length this again might not be a good approximation But for that we have this cost function J which tells us whether or not we're in that regime When it does work it reduces the computational cost by potentially up to an order of magnitude So this is quite a significant saving and it's worth doing for big systems It's not relevant for this application But we also do have a variation where we can use it for extended systems which we call the pseudo fragment approach They're just the last thing about fragments in big DFT And the systems I'm interested in talking about today We have well-defined separable fragments So these are single molecules, but in case you have something more complicated like a biological system So this is a lack as enzyme We also have a couple of indicators which we've defined based on the density matrix Which allows us to determine whether or not part of the system could be treated as a separable fragment And we also then have a fragment bond order, which means we can do things like draw maps between fragments So this is quite useful for analysis if you have a system where it's maybe not so intuitive what the fragments should be So just to summarize how we're doing large systems So in big DFT we have a range of methods from the cubic scaling to the linear scaling to the fragments And then also some implicit solvent You can see here that there's quite big difference in computational cost that this is for Costs of CBP, this is the OLED molecule You can see particularly that the big difference between the linear scaling and the fragment approach Which is still linear scaling but much cheaper And just in case you're interested in how big we can go We recently managed to treat the COVID spike protein which has about 50,000 atoms On 16,000 cores in six hours for a single point calculation So we don't quite need to go to the 50,000 atom limit for the OLEDs But if we really need to then we can Okay, so now I want to talk about going beyond the ground state So a few years back we already implemented in big DFT constrained DFT Which for those of you who are not familiar with this approach The idea is that we can find the lowest energy state satisfying some particular constraint on the density In that case we're interested in the charge constraint So this is particularly useful for treating charge transfer excitations And it's quite appealing because the cost is of a similar order of magnitude to ground state calculations So it's relatively inexpensive So the idea here is that we just add an extra term To our energy functional So we have some Lagrange multiplier We have our density kernel that we already introduced We have some kind of weight matrix which defines where this charge should go And then we have the charge that we want to transfer So the aim with CDFT is that we should find the Lagrange multiplier Which gives us the required charge transfer And then we just need to define where this charge is going to go So in big DFT we use our allowed in like approach Because this is very easy implemented with our support functions And we also implement this within the fragment of framework We basically associate charge to a particular fragment So CDFT and the fragments work all together So we already applied this a few years back This is the example I showed right at the beginning So this is a more kind of traditional host guest OLED structure So this is a Rydian PPY3 So one of these Rydian based emitters in the CBP host And we looked at different properties So both the charge transfer integral So this is based on kind of fragment orbital approach And then the onsite energies using constrained DFT And here we were able to see both the kind of effect of the disorder in the transfer integral And then in the onsite energies we could also see the effect of the environment So this is kind of very nice to see that actually the Particularly the environment does make a big difference We have quite a lot of dispersion due to the disorder But this is charge transfer excitations only And I mentioned that for TADF we also want to do local excitations And so we can't really use traditional CDFT We could use time dependent DFT This is the most common approach But one of the problems with TDDFT is that If you're using a semi-local functional There's a big problem with treating charge transfer excitation So they're usually seriously underestimated We can get around that by using hybrid functionals Particularly range separated hybrid functionals And going even further Tuned range separated hybrid functionals Work quite well for TADFs But hybrid functions with large systems are very expensive Even more so if you have to also tune for the particular system So ideally we'd like to be able to use a semi-local functional Not least because we don't have hybrid functionals Available in linear scaling big DFT at the moment We could also use another approach called Delta SCF So this is the advantage of being cheap But you can have problems with local minima And also with spring contamination So this kind of motivates us to define a new flavor of CDFT Which we call transition-based CDFT And so the idea here I'm not going to go into detail in the mass But to give you a kind of flavor of what the constraint looks like We basically impose a constraint between orbitals So this could be in this example Between the HOMO and the LUMO of a given fragment So if it's a single molecule You're basically imposing a constraint Between the HOMO of that molecule And the LUMO of that molecule As I'll show, it doesn't have to be the HOMO and LUMO But just for example So this is already compatible with the fragment approach So we know we're going to be able to do large systems In principle, it should be able to do Both locally excited and charged transfer states I'll show you how that works in practice And we're also including self-consistency So we're going beyond linear response So the first kind of thing you need to do In CDFT is find this Lagrange multiplier And it's not always trivial It can be a bit expensive Take a few iterations to do But the advantage if we have a pure HOMO-LUMO Transition is that we can't transfer more than one electron So actually what we can do Is just set the Lagrange multiplier arbitrarily large And we never kind of surpass the correct number of electrons To transfer or to if we're working with spin or not So this means that we completely avoid the need To optimize the Lagrange multiplier And so really the cost basically is the same As the ground state Of course we might not always want To do a HOMO-LUMO transition More kind of importantly We might not want to have a transition Between just two orbitals We might have some combination of transitions There's also a question of how do you actually decide What transition to impose So in our case we're using TDDFT This might seem a bit counterintuitive To do TDDFT and then constrain DFT afterwards But the idea being that we could do TDDFT For a single molecule And then do the TCDFT calculation Using that constraint for a much larger system So we're fine with paying the price Of one TDDFT calculation for a small system And so just looking at the asine So this is kind of the transition breakdown That you get from TDDFT Predominantly HOMO-LUMO But you have some small contributions From the HOMO-1 to the LUMO-plus 1 HOMO-1 is 2 to plus 2 And so the way we do this with our approach Is to do a separate calculation For each transition constraint And then we sum up the density kernacles Using this transition breakdown waiting So the cost does increase In that you need to do an extra calculation For each transition you want to impose But you can still use just an arbitrary large VC And you can see that for a slightly larger contribution You get a slightly bigger difference Between the pure HOMO-LUMO and the mixed transition By the time you get down to smaller contributions This effect is quite small So we wanted to test how this works For both local and charged transfer excitations So we defined kind of a test set Which has these asines which are predominantly local Just using this kind of simple HOMO-LUMO spatial overlap As our metric for local versus charged transfer And then we also looked at a handful of OLED molecules Which go from kind of mixed charged transfer Local all the way down to kind of fully charged transfer And these are kind of predominantly pure excitations But the singlets do have some kind of small Extra transition contributions So first of all the asines So we use as our benchmark data CCSDT values So we find that if we compare the trends of the CCSDT With our TCDFT with PB and TDDFT with PBE So this is a local system We expect TDDFT with PB should work well Our TCDFT gets very very similar results to TDDFT And so kind of the highlight here Is that with a semi-local functional We're doing as well as TDDFT But cheaper and we're doing better But Delta SCF for kind of similar costs And not too much worse compared to hybrid functionals So with local excitations We were quite happy that it seems to work well For the OLED so these more charged transfer like We don't have any CCSDT or other kind of reference values These are quite large molecules So we compared to range separated TDDFT values Which are kind of tuned for the system This is kind of our best reference we had So here we find just as an example So this is the most charged transfer like molecule You can see this big failure of TDDFT with PBE Significantly underestimating And we don't have that problem with TCDFT So if we look at the errors We can see that actually with PBE We're outperforming both Delta SCF and TDDFT Whilst being cheaper as well So we're also kind of happy that we're doing Pretty well with the charge transfer excitations And if we just kind of want to look at this trend If we compare TCDFT and TDDFT with PBE We can see that for these local excitations We get very similar results So where TDDFT does well so do we And where TDDFT does badly We get quite different results And so just to kind of summarize We're pretty happy with the TCDFT approach At least for the molecules we've tested so far So we can do both the local excitations And the charge transfer excitations While you're still using a semi local functional With a cost which is not too much higher Than the ground state calculations So this should enable us to go To the big systems that we're interested in So just very briefly I want to wrap up By showing an example of how we might use this in practice So this is the TCZPM molecule that I showed earlier Which is kind of a prototypical TADF emitter So we started with a snapshot from classical molecular dynamics This 500 molecules If we want to look at first of all Just the disorder in the gas phase We used this cost function J that I mentioned earlier To kind of compare how similar the structures Each molecule are from this snapshot And that enabled us to identify some representative molecules Out of the snapshot to be able to calculate the parameters So in the first instance We've just looked at the ground state parameters So both the valence x-ray photo electron spectroscopy Which we compare with some experimental values And we saw kind of Some small influence of the disorder in the spectra And then we also separately did the core XPS Using a different wavelet based code Called madness and the delta SCF approach Where we also kind of saw some small influence of the distortions So this was interesting But really we want to be able to look at the excited states So TCZPM was one of the molecules in our tester Where things seem to work reasonably well And so what we're working on the moment Is the idea of doing this workflow That as I said we start with TDDFT for a single molecule We know in this case that both S1 and T1 are pure excitations We extract some molecules and look at the TCDFT gas phase disorder And then also we want to ultimately look at clusters And see how the excitation energies are affected by the environment And so the idea here is if we can bring all these things together This is going to give us an improved description of the single triplet splitting And ultimately hopefully help us to try and identify And predicts kind of more accurately Which molecules could be good TADF emitters In which kind of environments So just to summarize So I've kind of talked about how we do DFT with wavelets For very large systems going to the kind of tens of thousands Of atoms regime So this is either using linear scaling big T Or for systems where we have some repetition we can exploit So we can use the fragment approach to reduce this cost further I also talked about the TCDFT approach Which enables us to do locally excited and charged transfer states With a cost which is not too high a price to pay And which is compatible with our fragment approach And then just briefly about this TCZPN How we've looked at the ground state disorder And we're trying to understand the excited state in the moment So there's a few future directions So this idea of looking at the environmental effect of the explicit excited states We also haven't mentioned anything about forces These results for versical excitations We want to look at adiabatic And also kind of look at the dynamics of the excited states So that's something that's also a work in progress Over the last few years we've also worked a lot on Pi big DFT, our Python interface So we've defined a lot of workflows To try and make these tools easier to use So we have kind of published Jupiter notebooks Which do this kind of process of the TDDFT to the TCDFT approach And then also working at the same time on a higher throughput approach for TADF And so just to finish with my acknowledgment So Martina Stallow who's in the audience She was a postdoc in my group is now a fellow based here Krytaan Flapper who's a PhD student in my group Luigi Endovesi and William Dawson who worked a lot on big DFT And then Natalie Fernando and Anna Ragoutz did the XPS measurements So also just to acknowledge the funding and computer time I do have a PhD position in my group in case you know anyone who's interested To work on this topic And then I'll leave you with the references And thank you for your attention And thank you very much for the very nice talk Questions Go ahead Very nice talk Laura I was wondering about when you do these calculations On the embedded system So you have the molecule inside sort of many other molecules And you use the semi-local DFT Are you worried about I mean I assume that it doesn't fully capture the effect of screening from the other molecules Is that something you're worried about and thinking about how that could be included Yeah, that's a good question I mean so at the moment we just we can't do hybrid functionals of large systems So we have to hope it's not too important But that is something that's on our radar for future work So what we can do is do hybrid functionals in cubic DFT And it's actually easy to go from linear scaling to cubic So one thing that we're kind of working on is doing a linear scaling calculation Then like a single iteration with hybrid functionals in the cubic scaling So that's something to look at in the future, yeah Any other questions Yeah, I was actually going to ask about that a little bit along these lines But I was with the comparison for charge transfer excitations If you're able to use hybrids, I guess if you're able to use a range separated hybrid Presumably that would do better than I think you were comparing with PBE TD DFT, which would be a problem in those cases But but not necessarily for a range separated hybrid and you know with asymptotic exchange I wondered if how that would compare with the method this TCDFT method So the reference that we're using here was the range separated hybrid Yeah, so because that's kind of the best reference we have really for these systems So you so it does compare very well with that Yeah, so exactly you can see as you say the TDFT compares very badly But the the TDFT is much more kind of reasonable And similar kind of accuracy to the PV zero comparisons with TDFT So that went fast for me. Thanks Other questions I think I think we might be ready for coffee So let's thank Laura again for a really very nice talk And coffee see a coffee Okay. Hello everyone So I'm Dorma Corrugnaneza from University Catholic Lever in Belgium And I'm going to be chairing this session in which we have three speakers invited speakers Which is before that we'd like to have an announcement by Emini Otter Hello. Good morning. This is a specific announcement for the For the members of the scientific advisory board We have a meeting at lunchtime and we have a table in the cafeteria reserved for that So please remember to come to go there and it would be important to go swiftly The agenda is quite packed And the the lunch break is not very long And so a general announcement to everybody else Which is going to be something very un-British to say If you see us the members of scientific advisory board around in the queue We are going to try to jump the queue We hope that you will allow us to do so Otherwise we don't have the time to discuss the things we have to discuss. Thank you very much Thank you Emilio Okay, so the first speaker of the session is Michele Simoncelli And he's from the TCM group in Cambridge And he's going to speak about transit in the thermal sorry conductivity of solid The floor is yours Thank you Okay, so let me start with Let me start with an outline of the talk I will start discussing the Wigner transport equation And how it can be applied to describe thermal transport in materials ranging from crystals to glasses I will show how this formulation allowed to describe thermal transport beyond the semi-classical Piers-Boltzmann limit Encompassing the emergence of particle wave duality for heat Then I will show how this formulation can be exploited To understand how to exploit structural disorder to vary the magnitude and trend of the thermal conductivity The second part of the talk will focus on amorphous solids Particular I will describe the the recently developed RWT computational protocol to comprehensively describe Thermal transporting glasses from first principles I will discuss how this allows to describe the effect of an harmonicity on the high temperature thermal conductivity of glasses And then I will show some application of this protocol to silica glass amorphous carbon and amorphous alumina Let me start from very general trends The thermal conductivity of crystals display a decrease with temperature You can see here experiments Scatter points and theory The lines for two different crystals You can see that the high temperature scaling is proportional to t-minus one And this is explained correctly by the Piers-Boltzmann theory for crystals, which was developed long ago This equation describes thermal transport as a particle-like phenomenon Particular we have that the microscopic heat carriers Drift and scatter similarly or in an analogous way to the particles of a classical gas On the other hand, we have glasses which display a conductivity which is very low And display a very different trend so we can see here a slightly increased with temperature This different trend is related to a different heat transport mechanism Which was explained about 30 years ago by Allen and Feldman And in particular they Show that the heat transport in these materials is mediated by The coupling between quasi degenerate vibration like in states And this theory holds in the regime in which disorder is the limiting factor for heat conduction Actually, this picture is not complete because we have materials which are technically crystals But they display a conductivity which is very low and deviates significantly for the t-minus one scaling Observer in simple crystals such as silicon or silicon dioxide So here you can see that in these materials for example lantanum zirconate Or cesium lead bromide which are materials using thermal barrier coatings or thermoelectrics We have that pyres Boltzmann theory the solid line deviates significantly from the experiments So in order to understand what is the main difference between these crystals such as silicon and silicon dioxide and lantanum zirconate for example We have to look at the phonon dispersion relation In silicon we have that phonon dispersion is very simple We have a small number of phonon bands which are blue here And anharmonic line widths which are shaded red here are very small So we have interband spacings which are much larger than the line widths On the other hand if we look at lantanum zirconate at high temperature We have a lot of phonon bands and they are significantly overlapping due to large anharmonic line widths So now I will show how to describe the thermal properties of these materials beyond the pyres Boltzmann transport equation using the Wigner formulation that we discussed a few years ago The starting point of our work is the density matrix formulation in particular We start from the one body density matrix here q q second are wave vectors belonging to the brilliance zone of the crystal B and alpha are atomic and Cartesian indexes At equilibrium we have that the one body density matrix Is a diagonal in the wave vector this corresponds to the fact that at equilibrium temperature is homogeneous And we know that the density matrix of a system at equilibrium is diagonal in the basis of the against states of the Hamiltonian If we drive instead the system out of equilibrium applying for example a temperature gradient We have now that the one body density matrix acquires of diagonal elements in the wave vectors And this corresponds to the fact that atoms vibrate more where temperature is hot less where temperature is cold And in order to describe this we need basically wave packets and so more than one single wave vector The natural object to describe this out of equilibrium system is the Wigner distribution function Which is shown here you can see the expression is quite complex But the take-home message is that whenever you have a one body density matrix carrying off diagonal elements You have a beginner distribution which depends on space And this is particularly convenient to describe this out of equilibrium regime, which we are able to track Position which is related to the temperature gradient driving transport as well as the wave vector Which is a convenient label for the against states of our system Now this object is a matrix And in particular a matrix in the phonon band index is s and s prime And one can show that the diagonal elements of this matrix are actually equivalent to The phonon population appearing in the standard piers-bolzmann equation While the off diagonal elements describe something more And using the language of quantum optics we call them coherences Here you can see the transport equation for this Wigner distribution function You can see that you have a commutator reminiscent of a density matrix evolution And an anti commutator which is generalizing the term the drift term appearing in the piers-bolzmann equation This equation is useful because it allows us to compute the heat flux and therefore the thermal conductivity The final result is quite complex, but it has a very simple interpretation We have that the conductivity is the sum of two terms the first term in green here Is exactly the same that you obtain solving the standard piers-bolzmann equation and describe the particle like or intra band propagation of a vibrational wave packets The additional term which we call coherence conductivity Becomes relevant whenever you have the interband spacing which is smaller than the line width So that you have significant overlap between different bands And it is possible to show analytically that in the disordered harmonic limits this coherence conductivity becomes equivalent to the allen-enfellmann conductivity for glasses Then another work derived the same result using or let's say they derive a similar result relying on the green cubo formulation And later also with Giovanni Calderelli, Larambe and Fatto we show that Actually the two approaches are totally equivalent And very recently Fiorentino and Baroni extended this demonstration of the equivalence also to the hydrodynamic regime We can compute all the quantities entering in this thermal conductivity expression from first principles Here you can see the results from for Lantanum zirconate So in in green you have the population or piers-bolzmann conductivity Which features the t minus one scaling and in blue you have this additional coherence conductivity, which has an opposite trend And if you account for both of them you correctly describe the total conductivity At low temperature we can see that the particle like conductivity is dominant to the coherence conductivity And this is because we have interband spacings Larger than the line width while at high temperature we have interband spacing smaller than the line width and the coherence conductivity is dominant Now we want to understand more in detail what's happening And so we resolved the particle like conductivity and the coherence conductivity in terms of A density of particle like and coherence conductivity I will not enter into the details here But it is possible to show analytically that the relative strength between particle like and wave like conductivity Scales as the ratio between average Interband spacing and line width And this allows us to draw a regime diagram for thermal transport In which we have a frequency and line width here So here in the orange region we are in the overdamped regime in which line widths are larger than the frequency We cannot say anything about this regime using the Wigner formulation because we are relying on the fact that Phonons are well defined quasi particles However, if we go here in the particle like propagation regime, we have frequencies much larger than the line width And this corresponds to the Pires Boltzmann or semi-classical regime meaning that if we are in this Region we expect Pires Boltzmann theory to be accurate And then Increasing the line width we cross the average interband spacing and we end up in the wave like tunnel in regime Which the Wigner function Formalism is needed Now we can check computationally these analytical expectations And the most natural way to do that Is to basically recast this relationship in the space domain by just multiplying these by the group velocity One can show that the ratio between average interband spacing and line width Multiplied by the velocity gives you the ratio between the mean free path and the typical interatomic spacing So we computed the mean free paths of our vibrations at 200 kelvin So at the low temperature where the populations Conductivity is dominant compared to the wave like conductivity We can see here every circle is a phonon sampled in our thermal conductivity calculation The size is proportional to the contribution of this Vibrational mode to the thermal conductivity And the color quantifies the origin of the contribution. So we use green for particle like propagation blue for wave like tunneling and red for something which is in between We can see that the transition from the particle like to the wave like propagation is a non-sharp And it is centered around the Interatomic spacing However, here we are at low temperature in which particle like conduction is dominant If we increase temperature an armonicity kicks in and it gives Basically lower conductivity and Short-term in free paths. So now we have a lot of vibrations which are below the Interatomic spacing and we can see that all these vibrations are contributing to transport in a wave like Mander Now I will discuss how it is possible to Exploit structural disorder to actually tune the scaling law of the thermal conductivity of solids Here you can see alpha quartz and meteoritic 3d mites to materials which have the same chemical composition But different structural disorder. You can see primitive cell of quartz contains 9 atoms meteoritic 3d mites We have 72 atoms and the phonon dispersion relation you can see are quite different So the average energy level spacing of quartz is much larger than that of meteoritic 3d mites This material Is interesting because it can be found actually in meteorites Here you can see a picture of the steinbank meteorite which fell north 300 years ago And our collaborators at Sorbonne University managed to retrieve a sample of this material and perform Experiments on the vibrational and thermal properties of this material So let me start from vibrational properties of alpha alpha quartz, which is the Very well known a very well known material Here you can see the line width as a function of frequency for alpha quartz at different temperatures and We can see that vibrations are never in the overdamped regime, which is in purple here So we can use the vegana formulation to describe it And most importantly the line width are always below the average energy level spacing, which is the line here So we have that quartz is a simple crystal and we can see a signature of This being a simple crystal also in the raman spectrum We have a small number of raman active modes And we can see that at different temperatures the number of peaks in the raman spectrum is always the same So vibrational Again states are always well separated If we look at meteoritic 3d mites instead We have that the line width distribution is similar to that of quartz But we can see that it crosses the average energy level spacing Which is now smaller and Then if we look at the raman spectrum, we can see that if we go from low temperature to high temperature The number of peaks actually decreases because we have a merging of the raman peaks due to the fact that the harmonic line widths are increasing with temperature Here we computed the thermal conductivity of meteoritic 3d mites and of alpha quartz and compare them with the experiments experiments for quartz Taken from the literature. These four 3d mites were performed by our collaborators in paris group of etienne balane And we can see that we have a good agreement between theory and experiments in both cases Now we can expect the origin of this mild decay of the conductivity of meteoritic 3d mites significantly deviating from t minus 1 So here i show the population and the coherence conductivity dot this population dashed is a coherence And we can see that this mild decay Emerges from having a significant role to play by the coherence conductivity So this shows that meteoritic 3d mites is a complex crystal in which coherence conductivity is larger than population conductivity at high temperature While quartz is always a simple crystal We can study different polymers for example alpha crystal ballite Which has the same ring statistic of 3d mites and again We find that it is a simple crystal in which heat conduction is dominated by particle like conduction mechanism And finally we can go to the fully disordered case which is a vitreous silica And we obtain a conductivity which is again in good agreement with experiments in this temperature range And this plays a trend which is completely different from that of crystals So to sum up we have shown that in meteoritic 3d mites we can achieve a scaling of the conductivity Which is totally intermediate between the decreasing trend of quartz and the increasing trend of silica Now in the final part of this talk i will focus on The calculation of the thermal conductivities of amorphous materials In particular how to compute the conductivity of glasses using atomistic models containing hundreds of atoms So in order to understand the what are the subtile aspects related to the computation of the conductivity of Glasses using small models. We have to look at the line with distribution for these different polymers We can see that in general Structural disorder doesn't have a strong effect on the line with distribution But what changes is the average energy level spacing So in crystals average energy level spacing is well defined But if we look at glasses, actually we have that the average energy level spacing Depends on the size of the atomistic model that we use to represent our Structure our glass And there are some subtle aspects related to this to be more precise it is useful to start From the vigner thermal conductivity expression Here i've denoted with half the distribution which is A lorenzian That is basically determining the strength of the coupling between vibrational eigenstates having different energy So within the anharmonic theory this distribution is a lorenzian which has a finite Full width at half maximum determined by the line width And instead if we go to the harmonic limit in which an armonicity Phases out this reduces to a direct delta And long ago allenenfellmann Discussed how this direct delta must be broadened with a distribution Having a full width at half maximum of the order of the average energy level spacing to understand why This prescription we have to analyze the energy levels of a finite size model of a glass So we consider a finite size models containing n atoms and then we compute its energy levels Which are schematically represented here by the Horizontal lines we have that the average energy level spacing is the ratio between the maximum frequency divided by the number of modes and If we then look at what is the overlap between these different energy levels due to anharmonicity we have that At high temperature We have significant overlap between neighboring vibrational energy levels because anharmonic lines are large But if we reduce temperature We have that the line width becomes smaller and we no longer have significant overlap so using just the Bare anharmonic theory on this finite size model would imply that There is no coupling between these neighboring vibrational energy level This is actually a finite size effect because we can double the size of our model So that we doubled the number of vibrational energy levels and so the Average spacings becomes smaller and now we would have a still coupling between neighboring vibrational energy levels So this reasoning shows that actually In the harmonic limit and in the infinite size or astronomically larger Larger size limit. We would always have coupling between neighboring vibrational against states And therefore using the allen and felman theory Using a broadening of the order of the average energy level spacing ensure that this property is is This property is respected We have checked computationally what happens if we evaluate allen felman theory in finite size models using A calculation of the gamma point only Or using the Fourier interpolation to actually improve the Accuracy with which we sample the vibrational properties And here we can see that in the calculation Done using the interpolation We have that we have a range of value for the broadening parameter in which the conductivity And broadening so we refer to this as a convergence plateau for allen felman theory Instead if we go if we look at the calculation at the gamma point only We actually have that it it is very difficult to achieve computational convergence And these statements are Strengthened by these additional tests performed on a larger models containing 5000 atoms We did this calculation relying on the state-of-the-art machine learning potential Discussed by Erard and other in these mpj computational materials Here we can see that increasing the size of the model we actually obtain Convergence plateau also doing a calculation of the gamma point only and the The convergence plateau using the interpolation mesh is wider so this test shows that actually We cannot consider An harmonicity when the line widths are a smaller than the value at which the convergence plateau starts So we have actually to regularize our an harmonic theory in order to make sure That we respect the physical property of always having coupling between neighboring neighboring vibrational eigenstates in the low temperature limit How do we do this in practice Here you can see a calculation Avigner and harmonic calculations are using the Lorentian on the models having different size And using or not the Fourier interpolation mesh to extrapolate the bulk limit So the blue line here is a calculation done at the gamma point using a small model orange solid is a gamma calculation for a large model containing 5000 atoms Then dash blue is the small model I studied with the Fourier mesh to extrapolate the bulk limit and the dashed orange is the large model started with the Fourier Mesh for the bulk limit. We can see that in the high temperature regime. We have good agreement between all the calculations that are Studying a system which as a size of a thousand of atoms either explicitly starting it with a very very large Reference cell or extrapolating it extrapolating the bulk limit with the Fourier mesh If we go at low temperature, we can see that We have a Divergence of the thermal conductivity which occurs at a decreasingly lower temperature as we increase the size of the model And this is a finite size effect reminiscent of the crystal like divergence of the conductivity that we observe in crystals And we can get rid of this in practice by employing a regularization Based on the void profile. So we replace the distribution appearing in the thermal conductivity expression with a void profile Which is the convolution between a Gaussian and a Lorentzian and we use The eta parameter for this void distribution equal to the beginning of the convergence plateau showed in the previous slide So this prescription ensured that when we are at very very low temperature. We are computing the conductivity using the The dirac delta appearing in the allen and felman expression And as we go at higher and higher temperature, we are actually accounting for an armonis T using the Lorentzian distribution Here you can see that the black line is the regularize beginner conductivity and the Red is the allen felman Calculation so an harmonic effects in vitro silica are very weak And here you can see that we have good agreement between regularize the beginner conductivity and beginner conductivity in the large model So in order to understand why the effects of an armonicity are very weak in vitro silica We again look at the beginner conductivity expression And we see that in the high temperature limit The thermal conductivity is actually a Lorentzian weighted average of velocity operator elements The weights in this average depends on the frequency difference between of the difference in frequency between the eigenstates couple So increasing temperature we have that the anharmonic line with increases so velocity operator elements related to eigenstates with increasingly larger frequency difference becomes more and more important So we look at the velocity operator as a function of the frequency difference And the frequency average and we can see that in vitro silica. This is almost constant with respect to the frequency difference so basically This explains why in vitro silica we have weak effects of anharmonicity and also It explains how anharmonicity can more generally affect the thermal conductivity of glasses By looking at how the velocity operator behaves We can have we can understand what will be the effect of anharmonicity if the velocity operator were increasing with Frequency difference anharmonicity would increase the thermal conductivity On the other hand if it was decreasing with the frequency difference anharmonicity would have had decreased the conductivity Quick a quick comment on the application of this computational framework to other amorphous material We have applied it to amorphous carbon Different densities and again we managed to describe Quite quite accurately the experimental measurements We check carefully finite size effect also in these materials and The capability of these regularization protocol to actually achieve computational convergence using small models And the very final slide is about the application of this framework to amorphous alumina Which allowed us to study this material from first principle And again we managed to obtain a good agreement between theory and experiments and also study how the Amorphous alumina can be Synthetized at different densities and we studied how density affects the conductivity of amorphous alumina finding that Basically increasing density gives you an increase of the thermal conductivity curve And so I come to the conclusion of my talk We have shown that the beginner formalism allows To account for particle like and wave like heat conduction mechanism and to describe materials with different degree of disorder We have shown that vibrations Which have a mean free path below the Interatomic spacing actually contribute to transport due to their wave like capability to interfere and tunnel We have shown that structural disorder can be used to engineer the thermal conductivity of solids And finally, we have shown that these void regularization protocol allows to study Glasses from first principles Thank you very much, and they are open to questions Did you get any questions? Very interesting talk. I just I have a kind of a naive question, but just so I understand it. So if I guess what you're saying is that in in a disordered case As you increase the number of atoms per unit cell As you increase the number of modes You know, even if they have a fairly small line with Because there's so many modes, they're going to start to overlap In that case You expect the coherence or the you know, wave like conductivity to dominate or increase substantially. Yes, okay, so it's basically Yeah, so any, you know, kind of disordered system Is I kind of maybe missed the very beginning of your talk But that's kind of a general expectation that this wave like contribution would be very important for any disordered system Where you have a lot of modes almost independent of temperature because You'll have so many modes so densely Pat, you know spaced that even a small With line width would lead to overlap Yeah, indeed. That's correct The the point is like it depends on how strong the overlap is because you have basically different ways of Tuning the strength of this overlap One is a varying an harmonistic. So you vary basically the your line width Which can be varied for example with temperature and the other is very structural disorder because you can Basically engineer more or less the number of modes by choosing the number of atoms in your primitive cell And I have one follow-up question, you know, when you're when these modes are overlapping You know energetically It seems like tunneling is possible But I guess it also depends on the eigenvectors of the modes at some level I I maybe Could you imagine having modes in a disordered system that are spatially distinct not overlapping? Yeah, so that's a good point actually because having overlap I would say is Necessarily necessary but not sufficient condition to have a contribution. So you need overlap and also you need Non-zero velocity operator elements coupling them. Okay, that's kind of the So if you have a localized modes, for example, they would A localized mode would be a more a mode having zero velocity operator element with all the other modes Right, okay, and that would never contribute to the conduct. Thank you One last question sorry So since these are disorder things, of course one crucial thing is the disorder configuration So there can be different kinds of configurations So is there a way of doing any configuration averaging? I I guess you did it for a single configuration Of course as you are increasing the size of your cell you sample probably many configurations but I mean how exactly you tackled it or What's your comment on it? Yes, so that's another important point in practice We checked this by doing calculation on different models having the same size And we applied the same framework to for example five different models all having a size of 192 atoms And we always found the variation in the conductivity course always within five percent The details are reported in the paper Which is Okay, in the interest of time. Sorry. We thank Mikaela again for the nice talk And we move to the next speaker syris drier from stony brook university Uh, yeah, sounds like it's working So thanks for the introduction and I'd also like to thank the organizers for the invitation to this this really interesting, uh, conference Before I start I just like to acknowledge my collaborators in this work So they are david vanderbilt and shang ren from ruckers university max dangle from barcelona John benigni from the flat iron institute center for computational quantum physics and sinisa co from University of california riverside So I think we've seen a couple of talks so far About anharmonicity. So I'll be talking about non adiabaticity And so just as kind of a very brief reminder, I think we have a lot of experts in the audience To the adiabatic approximation that we usually make when we do electronic structure calculation So in an electronic structure calculation, we have nuclei and we have electrons And we we oftentimes Separate these in our calculation based on the fact that of course nuclei are much heavier We expect their dynamics to be slower. And so we make the borne oppenheimer approximation And there's a variety of different ways of of kind of saying what the the The borne oppenheimer approximation is and and the physics that it that it results in But uh, basically the idea is that if we if we think about any kind of nuclear displacements, for example, the phonon mode We assume that they are slow. We assume electrons are in their instantaneous ground states with uh with static displacements And and also that the the nuclei if we construct an effective Hamiltonian for the nuclei, they basically move in the potential energy surface of these electronic states So, um, what I'll be talking about are situations where the the borne oppenheimer approximation Breaks down or by basically going beyond this adiabatic approximation. We get some interesting phenomenon or interesting physics And the two uh, the two Examples that I'll give are uh based on my title in metals and in magnets So first I'll talk about how uh In in metals There's a concept of a non adiabatic borne effective charge That requires again going beyond this borne oppenheimer approximation And then in magnets how we also need to go beyond the adiabatic Limit to to accurately include time-aversal symmetry breaking in the the phonon sector So both of these cases based on lattice dynamics beyond the the non adiabatic regime So starting off uh with with metals So one of again one of the nice things about giving a talk to a electronic structure audience is that uh, Born effective charges of the the main topic of this part of my talk is something that I think is known to to many of you And there are many uh experts in the audience On these types of quantities So it's a quantity that we use a lot in in first principles calculations dfpt lattice dynamics, etc So the borne effective charges are a tensor There's a value for each of the sub lattices in the crystal And then there's some directional indices i and j And there's again a variety of ways of thinking about borne effective charges You can think about them as for example if you'd apply an electric field in a given direction to your to an ionic material Then you'll get some forces on the ions You can also write this in terms of uh, polarizations that are induced by an atomic displacement or equivalent equivalently Moments of the the charge density that are also induced by some ionic displacement And so the the borne effective charges they have units of charge But of course it's a tensor and it's defined as a dynamical charge So it's defined through these perturbations So it's not a real static charge that exists in your material But one of the interesting things is that there is Another connection between the borne effective charge and some static charge And that is that there is a charge neutrality condition for the for the borne effective charges So this was uh shown rigorously in this seminal paper by piccone and martin That if you sum over the sub lattices of the of a crystal The borne effect sum of the borne effective charges is equal to zero and this This comes from the acoustic sum rule as well as the the the charge the neutrality of real charge in your system So the reason that we are interested in borne effective charges is that they show up in a bunch of phenomenon in in materials Including a ferroelectric polarization Just the calculation of phonon dispersions Electron phonon scattering dielectric screening electromechanical coupling, etc But one of the things about all these phenomenon is that they are usually associated with insulating systems And so the question is what about in metals? So in in metals, do we have a a concept of borne effective charges? Are they zero non-zero? etc And so if we go back to our definition of the borne effective charges at first sight it seems like It's not applicable to metals So in metals we would screen any electric fields that we would try to apply to our system Any static electric fields? The polarization electrical polarization is not well defined in a metal and also any moments of a of perturb charge density or also screen So it appears that we that maybe the borne effective charges are not relevant in metallic systems However, uh, the caveat to those arguments that I those kind of naive arguments I gave on the last slide is that the total screening will of course only occur if we give time for the electrons to screen the the Any electric fields? And so if we go beyond the adiabatic approximation what we find is that there is actually a A well-defined notion of a borne effective charge in in conducting systems And so the the approach that we that we take to define this quantity is basically to start with a fully non adiabatic uh Quantity similar that looks like a borne effective charge And take the the zero frequency limit of this quantity and see if we get something well defined so, um, so this idea Has been done in the past, especially in the group of francesco mowry And the idea is basically we can define a response function That is frequency dependent that basically contains this this susceptibility And this susceptibility is a current response to some atomic displacement And the susceptibility is frequency dependent and in general depends on wave vector And then if we take the the Wave vector goes to zero limit First and then we take the frequency goes to zero limit of this quantity. That's what we define as our our borne effective charge And so the key question is Is this zero frequency limit of this response function something that is well defined something that is That can in general be be non zero And so one way that we can understand this limit is to to look at a similar response function That may be more familiar to to most people And that is replacing this susceptibility, which is the current current susceptibility and this gives us exactly Uh the Yeah, so this gives us exactly the optical conductivity And one thing that we we know from textbooks about the optical conductivity Is that if we take the zero frequency, uh limit um of of this this, uh, uh q equal q goes to zero Susceptibility then we get a a constant, which is the drudal weight And so we have a constant divided by frequency and so we get a divergence In the in the optical conductivity at low frequency And this is the well known drudal peak in the the real part of the Of the optical conductivity Now we see again in textbooks And so just as kind of a reminder that so the drudal weight we can write in a variety of ways I just write here an expression from independent particles But basically the physics is it gives the the density of free electrons that are available for conduction in in a material And so it's kind of a defining aspect of a metal a metal has a non-zero drudal General non-zero drudal weight and a insulator has has zero So, um, we just showed that there's this divergence in the optical conductivity So does that also occur in this other response function? And the answer is no at least if we have time reversal symmetry and the reason is because, um, the The the current is is a time reversal odd quantity Um, and so the the for the optical conductivity we have two odd quantities, which give us an even quantity overall Um, whereas the the lattice, uh, this atomic displacement Is a time reversal even so we have an even and an odd quantity It's odd overall and then what that means is that, um, because again in time reversal symmetric metals We can't have a steady current that is, uh, uh in response to a static displacement and therefore the, uh This susceptibility needs to go to zero as frequency goes to zero So, uh, kind of in a hand waving way, uh, we can say that this This we should get a well-defined, uh, uh, zero frequency limit for this quantity We can actually go one step further in comparing the optical conductivity and this other, uh, quantity Um, in order to determine what happens to that sum rule that I that I mentioned before So, um, if we we formally compare these two quantities We see that the the sub lattice sum of this kind of born effective charge finite frequency born effective charge Is is, uh, is related directly to the optical conductivity And therefore if we take the zero frequency limit of both sides of this equation, um, then we see that the the sub lattice sum of the born effective charges is, uh, actually basically, uh proportional to the druda weight And so that means that of course in all metals the sub lattice sum of the born effective charges will not vanish Um, and of course this reduces to the to the charge neutrality condition in insulators Where the druda weight is is equal to zero So the the physics of this the sum rule are actually pretty simple So the idea is that, um, again if we have our, uh, our crystal and now we we group the electrons into into Ions that are tightly so electrons that are tightly bound to the nuclei and free electrons And then, um, the sub lattice sum of the born effective charge basically means that we are shifting the entire crystal But if we shift it in a non adiabatic way, we will leave behind some charge density And this charge density is left behind is exactly the free charge which is given by the the the druda weight Okay, so, um, I've kind of just made some some kind of hand waving formal arguments But, um, we can actually calculate, uh, these quantities. Um, I won't go through the the details of the calculation, but we we use kind of modified density functional perturbation theory techniques And again in a very similar way that was it has been calculated before in the Maori group Um, requires electric field perturbations atomic displacement perturbations And we have to be careful of course because we're we're treating metals Um, so these calculations were, uh Implemented in the in the abonnet code and here I just flashed some of the computational details and uh, and specifically we use one a interpolation because we We need a lot of k points to converge the various quantities that are uh, that are necessary for these calculations So I'll just so show two, uh, two examples. One is a simple Simple metal, uh, FCC aluminum. The other is a, uh, a doped insulator So we'll look at electron doped cubic strontium titanate So for FCC aluminum, um, here what I'm showing the in blue we have the boron effective charge So it's clearly non-zero Um in in black here if you just focus on the black curve that is the Basically minus the the druid of weight and if we if we add those two quantities together Just focusing on the red curve we get, uh, the the sum rule satisfied to very High accuracy as long as we have enough k points to converge the the druid of weight and the the boron effective charges So the situation is similar for for, uh, uh, strontium titanate So what we're we're doing here is kind of the at least the thought experiment of doping strontium titanate across um, the whole conduction band manifold made up of, uh, the titanium T2g states and here we have boron effective charges for, uh For the oxygens and for the titanium and the strontium And we see that the boron effective charges change as we as we dope On the the left hand side of this plot these would just be the the normal adiabatic boron effective charges And if we sum up the boron effective charges over this whole range and we subtract the druid of weight then again we get this This sum rule to be to be satisfied Okay, so that's what I wanted to say about, uh, metals and this non adiabatic boron effective charges So that we find they're well defined in metals at least in certain cases where you have time reversal symmetry The sub lattice sum of the these boron effective charges is the druid of weight And we have this dfp t implementation for for calculating these quantities So now i'll switch gears a little bit again You'll see the theme of non adiabatic lattice dynamics cropping up, but now i'll be talking about, uh, magnetic systems So the systems with time reversal symmetry breaking And in this case i'll only be talking about insulators so okay, so The place that we that we kind of started when we were thinking about this situation is the place you usually start when you do a lot of Dynamics calculation you calculate the interatomic force constants So roughly speaking if you displace an atom you calculate the forces on all the other atoms construct the The force constants dynamical matrix get the phonons, etc So then the question is what happens in a magnetic system Well, if you turn on magnetism in your calculation, which I indicate by these little circles Then maybe you have magnetic moments on some of your atoms your your electronic structure your density changes and uh compared to The case where magnetism is turned off You have different forces and you have different, uh Uh different interatomic force constants Now if you reverse the magnetization Then, um, you know reverse the direction of these little, uh, uh circles Um, then the energy won't change so actually you'll get the same interatomic force constants and then your Uh your equations of motion that you get your phonons from will have time reversal symmetry Even though time reversal symmetry is broken in the electronic sector And so basically the idea is that the the interatomic force constant matrix is time reversal symmetry by construction And the reason is basically because we usually construct it by uh by calculating these static forces So if you want to explicitly include the uh time reversal symmetry breaking in the phonon sector that's present in the electron sector Then there's there's a variety of ways to do this So one way was originally, uh develop, uh or kind of originally written down by mead and truller in the in these uh in this seminal paper Um where we have to include in our effective Hamiltonian for the nuclei Not only the scalar potential that comes roughly speaking from the um the, uh The electronic energies at a given nuclear configuration, but we also have to include the, uh A vector potential and this vector potential is basically a a very Potential a very connection in real space. So here r is a is a displacement of atoms in real space And if you start from this effective, uh Hamiltonian and you calculate the equations of motion Then in your equations of motion you not only get the interatomic force constants But you get an extra term that is, uh, uh that depends on frequency So it's kind of a velocity dependence of the force And this extra term contains basically a berry curvature in real space. So again these rs are, uh displacements of atoms in real space And so we can solve this equation of motion we can get, uh, uh phonon frequencies And so, uh, we did this with the example of an insulating ferromagnet chromium triadide It's a 2d layered material. This is actually the bulk bulk calculations And uh, what we get is shown in this this kind of Complicated plot here basically for some of the phonon modes. So these are just phonon modes that are labeled by their irreps We have large Contributions to this g matrix this berry curvature And in those cases where this contribution, uh, or these these uh matrix elements are large Then we get large splittings of phonon modes. So here the left is just the interatomic force constants The right is what we call this mead trueller, uh, approach And what will happen is these these twofold degenerate modes will, uh, split into, uh, modes that have well-defined angular momentum So chiral phonon modes so we can list the angular momentum here And so we can also look at, um, the the physics of what is happening in this in this system And so basically what we have is we have magnetic moments on chromium But the strong spin orbit coupling is on the iodine atoms And so basically what is happening is if we have phonon displacements of iodine atoms Um, the the displacement of the iodine atoms is changing the easy axis for this, uh, for the chromium Spins and so the spins are canting and if we move in a in a closed loop Then the spins will sweep out some area And uh, and that will give us this non-trivial berry curvature So this is this is the physics behind what we're seeing but this this physics is is, uh, is troubling in the way that we've done the calculation So actually well in the way that we've done the calculation we we've started from the born Oppenheimer approximation So we're we're still in the in the adiabatic regime But we see that the the the important dynamics in the system are spin dynamics And a spin dynamics as opposed to maybe, um, uh, kind of the naive view of electron dynamics are not necessarily faster than the than the the phonon dynamics The spin dynamics depend on the um on the the magnon frequencies And the relevant magnon so, uh, sorry, I forgot to mention we're always looking at the zone center The relevant magnons and chromium tri iodide are Lower in frequency or around the same frequency as the relevant phonon modes in in chromium tri iodine So we can't we can't assume that, uh, these Uh, that the magnetic degrees of freedom are fast compared to the the ionic degrees of freedom and we need to really, uh, Uh, compute the spin and phonon dynamics on the same on the same footing And so we did this in chromium tri iodide with a simple model So in the in the interest of time, I won't go into into too many of the details But basically just treating the the spin semi-classically and treating the the, uh, these A single set of degenerate phonon modes um coupled to a, uh, so phonon modes coupled to to a single magnon And, uh, taking this Hamiltonian and solving the the equations of motion for these coupled, uh, phonon magnons And basically what we see if we do this is that the splittings between these chiral modes Are greatly reduced so that's the sp spin phonon column compared to the mead trueller column And the physics is basically that the spins can't move fast enough to to keep up with the with the phonon Uh, uh atomic displacements and so the spins are sweeping out a smaller area and therefore we have a smaller barrier curvature Um, and also it depends very sensitively on the magnon frequency versus the phonon frequency So we can get a large splitting if the two coincide and uh, and we get some large hybridization So, uh, yeah, so that's all I wanted to say about about magnets Um, we we uh, in general conventional calculations don't explicitly have this time of virtual symmetry breaking in the phonon sector Um, we can account for it with velocity dependent forces But in cases where the the the relevant dynamics are the spin dynamics Then we need to include phonons and spins on the same same footing And so if you're interested in in, uh, magnets, uh, or magnetic And and phonon properties then just a shout out to a poster By another collaborator, Asier Zaballo, who's a student of max stengel He has a poster on on basically External magnetic fields affecting vibrational properties in in materials Okay, so that's that's all I I wanted to say about uh, non adiabatic lattice dynamics and metals and magnets So thanks for your attention and I'll take take any questions Harris, thank you for respecting the timing questions Just a curiosity, um in your definition of the of the effective charges There is this proportionality with the sociopathic ability the response function So your charges have a complex part, right? So if we if we um, if we go to finite frequency, then they definitely have a complex part Yes, physically, what is the meaning? Yeah, that is a good question. Um, so we haven't we haven't we haven't thought about the the the finite frequency version That has a complex part Um Yeah, I would have I would have to think about that that some more Yeah, it may yeah, yeah, I don't want to get myself in trouble Saras, thank you for informing us and Teaching us about some of these really subtle things. I want to make a general comment. That's for the audience And it's it's embedded in what you said We fooled ourselves for many years and a lot of textbooks are just playing wrong and in in defining things like polarization and and effective charges that that what's really Physical what can be measured is currents and and uh, we've Just make things hard for ourselves by defining polarization and acting like it's a static property That's that's well-defined. Now. What's really measurable is changes in polarization charges moving around and so once you have that attitude and then Then it's that leads to this modern theory of polarization that involves berry phases is that you should have thought of it as currents all along And and it's adiabatic currents and an insulator and then it's all it's interesting other stuff in metals And so thank you. Great. Yeah, thanks for the comment So what happened to bad metals? like Do you think that you can apply these to bad metals where The single particle theory theory might not work So everything I showed is is single particle physics. So it's definitely an interesting question about generalizing the concept of born effective charges Uh, uh to to systems with interactions Yeah, so I guess I guess there is a there is a uh, uh, uh, a frequency Range in which all the physics to deal with born effective charges is is relevant So in in bad metals where you would expect kind of the scattering time to be to be quite short Um, this this may not apply because what we'd expect is if we have a scattering time That is, uh The the scattering is is faster than kind of the phone on frequencies Then the scattering will serve to to get us back into the adiabatic regime Um, so that's the comment about scattering not about correlations or correlated systems. Yeah Okay, uh, thank you Um, I'm wondering if there are effects of non adiabaticity also on systems where one typically applies Uh, the standard way of calculating born effective charges say semiconductors Um 35 semiconductors or so. Yeah, so so there definitely can be with very small gap semiconductors and francesco Maury's group has worked a little bit on this So if you have a very small gap semiconductor in principle, you still have a gap. You don't have a metal um, but then of course the the um, uh, you don't have this this nice breakdown of um, or this nice, uh, dichotomy in terms of of Staying in the in the instantaneous ground state because the phone on frequencies can be of order of the gap Does answer your question no, I have a question in the In the case that you consider with sort of interaction with Magnons and phonons, so in principle you you use the approach of the very curvature which is at If you want linearize The equation in terms of omega can come from instead the self-energy non-adiabatic self-energy And these are what was the approach that we use the alternative approach. Do you think that in that approach? So in principle that approach contain all the the response even the response to Magnons is these you think that you can get the same result so considering both magnons and phonons with this second approach, which is equivalent to yours in the case So if you linearize with omega If you if you the one we using in the paper on the trimer and So so you're you're asking about about this approach. Yeah, if this approach is equivalent if you consider the full Self-energy non-adiabatic self-energy with spin orbit interaction. Yeah. Yeah into these or not I I I think I think Yeah, so so there I this this there's a few simplifications that we that we have here That that I think still could could be Nailed down better and so one of them is just we're treating this the spins a semi classical And I think that that could be a difference between between having the full The full self-energy and also We there's so so this the the the the spin canting mechanism So that kind of just the the the true magnon phonon coupling Is is only you know one part of the of the response So you or one part of the the berry curvature So you would also have a berry curvature with thick spin and and phonon Displacement, you know moving phonons in a in a closed path And and then a kind of combination of these So so those those we've also calculated in this context and they they turn out to be fairly small, but they're still corrections so Yeah, I don't know if that answers your your question, but I I think that this is kind of a more simplified Yeah, maybe we can discuss more Okay, let's thank Saris again We can move to the next talk by Francesco Makeda from the University of La Sapienza di Roma Okay, so good morning everybody and Thanks for the kind introduction So I'm Francesco Makeda and I'm a postdoctoral researcher at the Institute Italian technology currently hosted at La Sapienza Rome I earned my PhD two years ago at the kings college london under the supervision of Dr. Nicola Bonini who as anticipated yesterday by Francesco sadly Passed away a few months ago at a very young age therefore the organizer gave me the possibility to take for a minute of our time in order to remember him and especially in this event where a lot of his colleagues and friends are present and In a session also where most of the things that you are saying were of interest of Nicola research topic in order to remember him I will just Show some of these of the works that I think are very well known in the community and In the regard vibrational properties the study of vibrational properties of system for example, Nicola and the workers were The the the first one to pioneer the study of acoustic phonon lifetimes and thermal transport in freestanding strain graphene in particular they showed that in the long wave long wavelength limit the acoustic phonons acquire a finite line width therefore making them ill-defined ill-defined Ill-defined quasi-particles for samples whose dimension is larger than one micrometer And this is entirely due to the behavior to the quadratic dispersion of out-of-plane Flexural mode which are typical of two dimensional membranes and the And the apparent pilot this apparent paradox is solved by the introduction of a small amount of strain Which rectifies the dispersion of the phonons around the origin Another work which I think is very well known in the community and it's very interesting is the study the first example of first principle study of phonon phonon harmonicity anharmonicity is in graphite and graphene in which nicola and co-workers Computed the line width and the line shift of phonon as a function of temperature complete revenue by computing the third and fourth order force constants And as regards instead for example the Vibrational properties coupled to electronic ones One work which I find really interesting is the one in which nicola and co-workers study the intrinsic electrical resistivity of graphene And this was the first example of a fully ab initio calculation of the resistivity in which They were able to reproduce the low temperature resistivity both in value and temperature dependence of intrinsic graphene Whereas at room temperature There is quite an astonishing result, which is that the ab initio calculation Failed to reproduce the The value of the resistivity of around 40 percent and this is still an open problem and In the ab initio study of the graphene resistivity And at last I would like to remember also the work I did with him, which was mostly about electronic and magneto Magneto transport phenomena Indoped semiconductors Basically, we applied for the first time Ab initio techniques to the solution of the botsman equation in presence of magnetic field And we were able to compute Coefficients such as the old scatter in fact or silver coefficient Magneto Magneto silver coefficient and magneto resistivity all from first principles And a few words Shall be Chosen and it's difficult to choose the word in order to describe the loss that we all had Therefore, I will just say that he was A great master and a very kind person And I would gently ask you to hold one minute of silence in his honor while I show some Obvious personal pictures So that's nicola and Okay, so bust into the talk. So I will talk about the fetish charges Mostly in line. We've also already Presented in the very nice talk before mine and in particular we talk about the fetish charges and their effect on electron phone interactions Phonon dispersions and the infrared spectroscopy in phonon endopsomy conductors metals And hydrogen-based I temper to superconductors Naturally, all this work wouldn't Have been possible without the help of many collaborators among which Especially I thank Williamo Marchese, Matteo Calandra, Tibo Soye, Paolo Baroni and Francesco Mauri and so As already said in the talk before mine Effective charges are the basic ingredients which Is needed in order to describe the coupling of The vibrational system to the electromagnetic radiation In particular therefore every experiment which Produces a disturbance of electromagnetic nature In a crystal as to deal with the Detection of effective charges So let me start From the experimental side this time and let's consider two of the main Experiments which are used especially to study cutting edge materials Which may be made of small samples in order to characterize their vibrational properties and These two are the electron energy loss petroscopy the ills in which a beam of monochromatic electron impinges On the material and through the loss function of the particle one can reconstruct The excitation of the system in particular we are interested in the one in the infrared spectrum And also the infrared absorption of a synchrotronic light So for ills As I said the one is interested to to represent in a plot As a function of the momentum the energy loss of the particles in particular one can reconstruct from the loss of of the heat as a function of a momentum transfer height map Which can be superimposed over for example the phonon dispersion of the material if you are interested in the dispersion of phonon Like excitation this would be done also for example for poreronic stuff And for poreronic physics, but for what you are interested in this talk We will talk about most phonons and I will represent here for Graphite and bulk habn the experimental result of the reconstruction of the energy loss As a function of momentum Below there is the we represent the theoretical calculation that can Be Can are done through the calculation of stokes cross cross section And in particular the the thing that they wanted you to notice is that some part of the of the dispersion Are hot and some part of the dispersion are cold in the sense that the coupling of the electrons Interaction to the to the phonons is governed by something which has some selection rules And in particular these quantities are set before the the effective charges Interestingly this can also be be seen in the infrared absorption For example, here I report an experiment done on calcium aluminium silicide Which is a bcs superconductor of a critical temperature of about four kelvin in which we can see the Reflectivity spectrum as a function of the frequency in particular. We can notice that before the sharp drop due to the onset of the plasma frequency We have a very high reflectivity But which is not exactly one and therefore it allows for the appearance of some additional peaks that have room to appear And in particular here there was a detection of a peak at 200 centimeters to the minus one Which may be used to probe the infrared vibrational spectrum of Reflectivity which is related to the presence of phonons Of course, even even here the intensity Of the peaks is related to effective charges That are present also metals as was presented in the talk in the previous talk Now we would like to describe all of these Experimental features and have a general theoretical framework in which we can describe the coupling of electromagnetic radiation to to phonons in order to do so the object of desire for the theoretical description is usually The electric function the electric function which relates the external potential To the total potential that is present in the crystal We may make some Idealizations and some approximation in which we treat the system as an infinite crystal and we disregard the retardation effects of light So in particular we consider the absence of magnetic fields in such case all the phases may be contained in the Fourier transform of the The electric function which depends on the wave vector q And and on the reciprocal lattice vector g and g prime A useful factorization that can be used in order to study the the response is the one in which we separate the electronic And the vibrational degrees of freedom of the system Though this separation doesn't Doesn't mean that the two parts are independent and in particular the vibrational part is strongly Tied to the to the electronic one Uh, this separation comes from the many body from a many body point of view from the separation of the Coulomb kernel In an electronic and phononic contribution We are mostly interested in this part because it's the one which Which experiment that looks at the vibration probes and in particular it can be shown that this part may be rewritten as A function of quantities that are called Transverse effective charges, but we will call also them unscreened effective charges for reasons that I will show you later and also as a function of the macroscopic inverse the electric function of the electronic system and Beyond beyond the definition of the transverse effective charges We can also define the longitudinal effective charges, which Includes also the effect of screening and are defined as the product of the transverse or a screened effective charges and the macroscopic The electric screening of electrons and they can be operatively defined as the induced density Following a perturbation of the system plus the perturbation itself divided by the Magnitude of the wave vector This the difference between longitudinal and transverse effective charges is basically the fact that The is related to the presence of electronic screening It can be shown that transverse effective charges may be expressed as a function of born effective charges and dynamical quadruples and so forth and so on as far as one I can go with this function And the important thing to relate with experiments is that the is cross-section that I showed at the beginning Are related to the longitudinal effective charges, whereas the infrared absorption is related to the transverse one So we'll keep for now a static description for the first part of the talk whereas we will move to a frequency dependent one for the second part So as also anticipated in the in the previous talk What one is interested usually is to extend the definition of effective charges to the case of the optomy conductors and metals because What they told what they told up to now is more or less test book physics range in a different way But the thing is that the polarization seemingly is not well defined when free carriers are present. So we may ask if Even the presence of Of free carriers if that's where the electric tensor is strongly dependent on intraband terms And it's divergent on the tomas fermi scale. We can still define transverse effective charges We are also interested in understanding what what happens to observables which are related to the presence of macroscopic charges in the system So in particular, we will study the effect of this expansion of the formalism On the longitudinal or screened effective charges through their effects on the electron phonon and phonon dispersion in top semiconductors whereas for the Unscreened one we will study this Disastention with the effects on the infrared reflectivity spectrum So let's start from from the screen the effective charges now. Let's take a cubic silicon carbide, which is a Element semi conductors and let's dope it with A small quantity of of dopants of in particular of holes in this case And let's consider Effective charges and their definition So we can plot the screen the effective charges as a function of the wave vector along a high symmetry line in the brillouin zone And we can plot it for the undoped case represented by circles and for the doped case represented by squares We can see that the in the undoped case the screened effective charges tends to a constant value Which is the same one that can be computed by Standard codes such as quantum express at q equals zero Whereas for the unscreened case We have that the charges gently go to zero with The wave while the wave wave vector goes to zero This is mostly due to the effect of doping and the important thing is that for small doping This effect may be entirely Entirely is entirely contained in the in in the behavior of the of the screening Whereas the unscreened effective charges are mostly The same as as for the undoped case But this is not always the case when the doping increase and we get to a metallic situation The unscreened effective charges have a very strong Dependence on the the doping level even in the static case So from from from from the study of effective charges We can also study the the impact on those servables for example on the well known Effects on the leotio splitting in the textbook leotio splitting in order to do so we have to rewrite The dynamical matrix and the electron funnel coupling in a way in which they are proportional explicitly to the the electric Matrix in particular to the electronic part of the electric matrix with a bit of math math and a bit of Complications one can Reduce the formula for the long-range part of the dynamical matrix and of the electron funnel coupling as a function of electric of effective charges alone Therefore we can say that the effective charges are the quantity that Lead the the physics of effects such as the frolic coupling and the leotio splitting For the undoped case. This is naturally known for the doped case This is an extension For example, we can we can see from the The effects on the frequency of the electron funnel coupling which follow from the behavior of the effective charges And in particular we can see that the leotio splitting which is Tending to a constant in the in the undoped case Gently go to zero in the case of of of adopt semiconductors and the same can be seen for the frolic coupling Which should be divergent in 3d material in the undoped case. Whereas it goes to zero in the case of The presence of doping We have studied these both via ab initio calculations and also via vignette interpolation And we even in this case we we Ended up with the conclusion that the effect of screening screening are mostly determined By the the electron screening function in the weak doping regime Whereas in the strong doping regime also the value of the effective charges change in particular This is this is important Because the effect on those servables may be Seen for example on the line width of electron. We can plot the line width of the electrons as a dash region around the the electronic dispersion In the case in which we do not consider the effect of Free carrier the presence of the carriers in the material in use state of the art methods for the undoped semiconductors And we can plot the same thing where we introduce the effect of free carriers and we can see that the lines are strongly different Of course, this is not the whole story because one should have Should include dynamical effects in the screening But this was just to show that we can extend the The concept of effective charges in the static case to the presence of free carriers And they may display an important dependence on On free carriers presence Now for the for the second part of the talk, let's move to the realm of frequency dependence So up to now we just started only we just started only static charges and again a stress that we showed that we can define We can deduce and compute well defined unscreened effective charges even in presence of doping and In presence of free carriers also as in a metal And we can study in this case for example the reflectivity spectrum of Materials which are of technological interest in this case We can plot the the reflectivity spectrum of of h3s In which if we just use the drood model simple drood model, we have a drop. We have a sharp drop at the home plasma the plasma frequency whereas In general we consider the effect of scattering. We have Reduction of the reflectivity from one to a value, which is not very distant from one But is enough it's distant enough in order to Let Some additional peaks to appear which are the peaks that we are interested in because they reveal The presence of vibration in the system in this case the reflectivity is written as a function of the electric function 2 equals 0 so Is a function of of omega And we will we will our our interest is very Is very direct towards the study of better reflecting metals Which are the ones in which the reflectivity is not perfectly one One of these these cases are bad metals where there is a strong level of interactions Now as as Anticipated before for the epsilon to the minus one also for the epsilon we can perform a separation of the degrees of freedom in which we can We can isolate the the contribution of the vibrations to the to the To the dielectric tensor in particular we have a dependence on Quantities that are oscillator called oscillator strengths in which depend on unscreened effective charges That are the one that were the main The main subject of the previous talk in which These quantities may be seen as a Dynamical effective charges, which may be expressed as a bubble in which one vertex is a velocity The other vertex is an electron phonon coupling and then we have the electronic bubble As shown before these are well defined quantities for metals And now we found ourselves though with two definition of a screen effective charges One which follow from the static theory and one which follows from the dynamical theory Now the big question is which one shall we use in order to reproduce Experimental data and in order to for example reproduce The data of an infrared absorption Actually, we should use both in fact Let me introduce some physical scales and some Which are the frequency the momentum the line width and the typical interband frequencies which Which Which regards in interband transition? And let's consider for example the response due to the electronic bubble And then therefore to the the electric screening which appears Here in the bubble of in So basically the f minus f Part of the of the formula. So in this case we can have in this plot in the frequency momentum plot We have some regions which are related to interband transition and some regions which are related to interband transition particularly this is the eletronol continuum which Which is typical of of for example, so a lindard model And let's consider two Mathematical limits which are the dynamic and the static limit The static limit is defined as the one in which the frequency is sent to zero before the the momentum And the dynamic one is the one with the reverse order in particular Let's consider them in the clean regime. It has in the regime where the line width is zero So these two regimes are the regimes where we actually can do calculation because in the f t We are in this regime. We are in the clean regime. So we have these two limits at end that define the two Kind of effective charges the static and the dynamical one we know that for basically The response for for for the theory from the theory the response these two limits do not commute in the clean regime and the reason is that there is The arisal of an intraband term Which can be cured This discontinuity as soon as we introduce a bit of of dirtiness in the system and with dirtiness I mean the fact that the line width is is different from zero another The opposite of this regime is the one where we have an ultra dirty regime Where the where the line width is very much larger than the frequency, but we keep it still Less than the typical interband transition energy in this case the static and dynamical limits commute and in this case as discussed a bit in the Question session of the previous talk In this limit the evolution is so slow that the system is practically Considered to be evolving adiabatically So in this sense what we can say is that we have a mathematical description of quantities which is We can assess the mathematical description of the dynamical limit in the clean limit With which we can define and compute the dynamical effective charges And these limits correspond exactly to the physical dynamical limit Instead the mathematical description the mathematical limit In which we in which we can compute the static effective charges Which is the clean limit and in the static in the static regime It corresponds to the physical case where we have The system is very dirty. It's a bad metal and the the line width is very large A confirmation of this Of this physical argument may be found in the extended drud model in which one can express the The effective charge at a generic frequency at a generic line width At the sum of the dynamical effective charges plus the difference between the static and the dynamical effective charges Times a factor which depends on the ratio between the frequency and the line width This is a an interband an interband term which can be turned on basically by the dirtiness Uh in we can therefore call the What what what corresponds to the mathematical Clean static calculation of a quantity the adiabatic limit in which we In which we simulate the physical limit the physical limit of ultra dirtiness So let's see what are the consequences of this for example on the study of the reflutivity spectrum of h3s at 150 giga pascal at room temperature We can compute the reflutivity through the epsilon of omega tensor With a formula shown before Introducing in it the the effective charges computed in the adiabatic Static or static if you want regime The one in the dynamic regime and the one That are computed in the intermediate case with this formula This interpolating formula inserting for the line width and the frequencies The one that can be that are deduced from a shock calculation A self-consistent stochastic harmonic approximation And In this case we can see that the reflutivity spectrum of the adiabatic case in yellow and the one of the dynamical case in Light blue are very different and the reason is that Static and dynamical charges may be very different up to the The difference may be up to the order of 100 percent there in The intermediate regime instead is in between the two And from a qualitative point of view it will represent the experiments that are reported here on arbitrary scale as last as a last Bono slide. So before the in the in the previous talk We saw that there are some rules for the effective charges In particular the dynamical one, which may be related to the plasma frequency Also in the for the calculation of adiabatic adiabatic effective charges for metals we found that there is A Some rule which includes a kinetic term, which is dependent on the the expectation value of BMP And the term which is depend on the electron phonon coupling both terms depend on the derivative of the fermi function therefore they are strongly Dependent on the on the fermi surface and this formula Goes back to zero in the case of semiconductors, which is required by Charge conservation while this is not required by charge conservation in metal Because the charge conservation would be not on the on screen the effective charges But on the screened one and basically the dielectric screening kills everything and allows for Charge conservation In particular we tested this this this sum rule by first computing the the effective charges through through the to the calculation of the induced density by an external perturbation and computed the various terms of the expansion and Comparing it to the analytical formula that we found and we found that Direct calculation and the sum rule agree within a 1% error Therefore we are very happy of this sanity check And also importantly we did not do it only on h3s But also for example on test case on the test case of aluminum that where we find that the sum rule for the diabetic charges 6 Around 6 whereas the one for dynamical charges that was shown In the previous in the previous talk is around 2 So in conclusion what I wanted the message that I wanted to convey is that we can define screened And unscreened effective charges for the metallic case or in general in presence of Free carriers even for dobsome conductors The screened one are an important one in order to describe these experiments electron phonon While the unscreened one are important for infrared absorption We can assess through the ft calculation the clean static and the dynamical Regimes for the determination of the unscreened charges by direct computation But in order to reproduce experiment the two of them interplay in the general case And the independent independence on the of the physical limit that we want to describe We have to choose which one to use The second part of the talk I'm very in depth to guillermo marquesa for all the theoretical developments and And the and the calculation and therefore I refer to his post his poster in order to Have more details about the matter And finally I leave some bibliography for interested people and That's the conclusion of my talk and I open to questions questions Very interesting talk. So I had a question on the the the The beginning part where you calculate the the lotio splitting so I guess in in that case you had the static charges. So the the frequency was brought to zero and then And then q is brought to zero. So the the the effective charges were zero at q equals zero And I was just wondering are there is there a situation where You would have finite lotio splitting at q equals zero Yeah, yeah, that's that's yeah, so I I said that I was limited to the static case in the first part of the visitation in general one Of course one should consider the physics of the Of the frequency dependence of the Of the the electrics response In particular one should look at in the plane where there is the presence of the plasmons the plasma frequency, etc Etc as a function of that one should determine the region in which the screening is mostly static or Is mostly dynamic now this was done in order to show that we can still In the in the in the limit of high dope semiconductors where the the static case prevails We can reproduce the the the reduction of lotio splitting which is seen in some experiments And and we can still reproduce we can still define effective charges But of course these effects are strongly dependent on the region of the on the the omega q region in which you are and actually we are studying it in order also to understand which are the effects of these Consideration then for example in transport, so Yeah, it's it's it's very difficult The questions If not an interesting interest of time, I think we can thank again francesco So can I can we ask you to wait one minute for the scientific members so scientific advisory board to leave and not to overcome you Thank you I Hello, everyone. So we need to start the session a few minutes before 2 o'clock so we have two, we have a special thing that Michele Cusula will do for for Remembrance and then he will give us a talk on Monte Carlo Methods. He's an expert and and we're really lucky to have him talking about the Monte Carlo Methods and in addition he's from Trieste so one of the very few people here who actually is from Trieste. Michele. So thanks Richard for this nice introduction and thanks to the organizers for giving me the opportunity to present my work. The work is about structural properties and phase transitions of hydrogen and hydrogen-rich compounds using quantum Monte Carlo. So I'd like to dedicate this talk to Sandro Sorella who sadly passed away last year and he was inspirational for the whole Commence Matter community, for our community. That's a picture of him in one of the nice Telluride meetings in Colorado and the material I'm gonna show you today for remembering him comes from Matteo Calandra as well Giuseppe Carleo, Kozuki Nakano and myself. So Sandro trained and built a new generation of scientists by counting the number of PhD students and postdocs. We counted 27 of them mentored and advised by him and the majority of them found a permanent position either in academia or in important companies and those who have no position yet maybe are too young to get one. So and it's impressive also the widespread of his impact in terms of mentoring and building this new generation of scientists. So I've been lucky of being advised by him for my PhD thesis. This happened starting from 2001 and I remember when I met him his way of convincing me to work with him was saying let's do it on a lattice and that's a typical way of talking by Sandro using his own jargon, it was very typical of him and then I understood that by this let's do on a lattice it meant to develop a new Diffusion Monte Carlo algorithm by using a lattice regularization and that was the birth of a new method called now LRDMC. So his quest in terms of scientific research was headed towards efficiency and accuracy of the methods to solve key problems and one of the key problems is superconductivity in Stronnick or the systems and if you like there's a few rules of his career he started working on superconductivity back in early 2000 and then he kept working on that till the last weeks of his life and this is a paper posted to the archives modified a few weeks after his terrible accident where he provides the most advanced phase diagram of the dimensional hubber model in the interaction strength and doping. So if I'd like to highlight Sandro's gems I would pick four of them the first one is the stochastic reconfiguration algorithm used to optimize wave functions and thanks to this algorithm he proposed several new wave function and variational wave functions to describe stronger relation and then I'd like also to highlight his work on the algorithm differentiation applied in quantum Monte Carlo and finally the creation and the development of the two RVB code so I'm gonna go quickly through these points. The stochastic reconfiguration has been a pioneering work written by Sandro in 1998 and inspired several other works not only in quantum Monte Carlo but also in machine learning so there is a machine learning version of it called natural gradient descent which has been written contemporarily but the link has been discovered recently and the stochastic reconfiguration method is one of the best optimizers for any variational states ranging from just reform to neural network states and also it's used by optimizing the parameterization of quantum circuits as we have seen yesterday by the next talk of Tomahira. So his eternal quest for a better variational state is never ending right because you can always imagine of a better one started by exploiting at most the flexibility of the resonating valence bond. He was really a fan of Anderson and he liked the variational approach behind the RVB theory applied to strongly prosthetic spin systems but not only then he used that same framework to compute the pairing symmetry in cup rates and iron base superconductors by determining the symmetry of the condensates entirely from first principles in a strongly correlated framework like quantum Monte Carlo that was a very nice very nice work and also he proposed very recently a new paradigm for the chemical bond in case molecules possess local spin moments by coupling covalent bond with spin fluctuations within the Fafian approach that's I see as a new paradigm for quantum chemistry and let me mention about algorithm differentiation has been a pioneered the use of algorithm differentiation in the context of quantum Monte Carlo together with the Luca Capriotti and these lead to the led to the possibility of computing ionic forces in quantum Monte Carlo with the same scaling as as energy and so this was a breakthrough and open paved the way to performing qmc driven molecular dynamics and qmc phonon calculations within a correlated framework and finally I'd like to mention the the toolbar will be a code so the development started 22 years ago and and it's a it's a huge code it's an independent package very flexible can be used both on open boundary conditions and periodic systems with different variational functions and interfaces with its own DFT engine and there is a list then of family of codes which are interfaced to be to make it to make the easy it's huge as its usage easier and so I'd like to remember him as someone still biking with us and he opened for us he paved the way thanks to his findings thanks to his tools and we can keep biking on this road and with many things ahead thanks also to particularly to his precious work and personally I remember him as a very honest person he's a model for me of scientific integrity honesty passion so he was extremely passionate of his work and also was a hard worker beside of course the brightness of his mind okay so now let's move to these okay so let's now move to the content of my talk so that's the outline I will start by talking about the challenges behind materials containing hydrogen I'd like to talk about the high pressure hydrogen phase diagram if I have time I'll talk also about the proton transfer in water clusters and then I'll draw the perspectives so quantum materials so hydrogen and hydrogen rich family materials belong to the family of quantum materials because both electrons and nuclei are quantum particles all into the light mass of hydrogen and so it's quantum localization so the nuclear quantum effects lead to remarkable properties one of these properties is the high temperature superconductivity so H2H3S discovered and measured in 2015 was a record-breaking measurement in terms of critical temperature breaking the previous record of mercury-based cuprates and then it is followed also by other compounds like lanthanum hydride showing even higher critical temperatures so in this in this talk I'll focus on pristine hydrogen so hydrogen as we proposed I'm sorry for this misalignment of the PDF so hydrogen is proposed by Ashcroft that's supposed to be the paper by Ashcroft the Pirelline 68 where he proposed hydrogen as a room temperature of very high temperature superconductors because he could meet in principle the three golden rules of the BCS gap to maximize the BCS gap namely a large nuclear vibration a large electron phonon coupling v and possibly a high electronic density n now the electronic density n is the weak point of the story because hydrogen at ambient conditions is an insulator due to his molecular nature and is an extremely complex phase diagram the age of Ashcroft this was not known very well and actually the insulating region of hydrogen extends up to very high pressures and and so the idea is now to to make it atomic such that you lose the insulating properties borrowed from the the large homo gap of the molecules so and so the quest for the atomic hydrogen metallic has been the holy grail in high-pressure physics for for since many years and still it's you know a terra inconecta here because due to the very high pressure now expected for this phase now if you look at the phase diagram of hydrogen can be the complexity can be compared to to the one of all other compounds like water and they are shared by the if you like the similar situation of being quantum materials in the sense that you have the presence of a stronger interplay in competition between internal electronic degrees of freedom and the vibrational energies due to to the light mass of hydrogen in both compounds so what is the challenge here the challenge that has this these phases these crystalline symmetries differ by by very few milli electron volt per unit you need an extremely accurate evaluation of the electronic internal energies and also a very accurate treatment of the nuclear degrees of freedom by a quantum description and so on the top of that you should also be able to describe the interplay between these two ingredients so for the electronic part the quantum Monte Carlo method is a natural choice because you need to describe these phases over a very large range of pressure and temperature and thermodynamic conditions so the flexibility and accuracy of the crucial Monte Carlo in particular is requested in this in this framework and indeed the quantum Monte Carlo has been the method used to take all the phase diagram of hydrogen since the beginning and I'm happy that Richard is chairing this session because one was on the pioneers of studying the hydrogen phase diagram using quantum Monte Carlo methods with that is separately here I show the one of the the most advanced and well established phase diagrams for hydrogen using quantum Monte Carlo one published both published in 2015 one by Drummond in the intermediate range pressure range and the other by by Morales and co-workers in the extreme high pressure range so in this work I focus on on four main phases I'm just addressing the extreme high pressure range looking for the atomic the atomization of hydrogen possibly so I take into account the c2c which is the well established phase three and then the three other competing phases with respect to c2c in in this range namely cmc12 which is still molecular cmc4 which is again molecular and finally the cs4 which is atomic now let's have a look at the state of the art by Morales so you have this the phase diagram the enthalpy as functional pressure computed with static lattice so the the configuration the ionic geometry is fixed at the at the equilibrium configuration for dft df on the top you do quantum Monte Carlo and with quantum Monte Carlo you get this phase diagram you see that the atomic the atomic phase is very high in pressure above 650 gigapascal now if you include the lattice dynamics at the harmonic level you get a drastic change first of all the atomic phase is pushed down in pressure and secondly there is a point here where there are basically many phases competing actually all all of the three phases I mentioned before competing in a very close range in pressure and given the accuracy of these data points it's it's hard to discriminate between it's hard telling who's the winner here in this region so we'd like to investigate this and by the way at these these magic crossing point in pressures there is a transition detected experimentally and that's detected by by lubeir very recently 2020 where he found a drop in the optical gap signature of a phase transition and if you will look at the lower pressure side of the phase transition we have a good agreement between the optical gap computer with quantum Monte Carlo by David Saperli Carlo Pelleoni, Oldsman and Gorelov you have you have a pressure dependence which follows very well the experimental data but the higher pressure side of the phase diagram is still unclear so we'd like to address this part here now and to address this part here better than the state of the art we we included quantum anharmonicity so we need to go beyond the harmonic theory for the lattice dynamics due to the strong fluctuations of of the atom of the hydrogen nuclei and how we do that so we combined the self-consistent harmonic approximation computed at the DFT belief level with very accurate internal energies computed by diffusion Monte Carlo and we put together the the ska vibrational zero point energy because we are at zero temperature here in this work with the DMC internal energy for the finite phase diagram so I skip the description of the self-consistent harmonic approximation given very nicely by Lorenzo Monacelli yesterday and in terms of the quantum Monte Carlo technicalities we used a wave function written as a product of just a factor times as later determinant all of all both both factors developed on a Gaussian basis set periodized and we the just with one body two body and many body parts to reduce the variance so we've been very careful in doing quantum Monte Carlo in this case because we need a very accurate extrapolation to the thermodynamic limit so we simulated sizes up to 1024 atoms for the atomic phase and up to 7068 atoms for the molecular phases and this this been done systematically for any for any crystalline symmetries and for any pressures studying this work and additionally we also we checked the impact of the fix another approximation because we carried out diffusion Monte Carlo on the top of the variational wave function so we analyze the impact of the fix another approximation bias by relaxing by optimizing the nodes at a smaller supercell size of 96 atoms and by correcting this by this bias also to the larger supercells okay so that's the the results we got if we compute quantum Monte Carlo on a static lattice so on the on the centroids on the deformed lattice by quantum unharmonicity provided by ska so it's not the equilibrium geometry of the ft it's the equilibrium geometry with including quantum and harmonicity and so that's the phase line with the static lattice now if let's focus now on the two on two phases the atomic one the season four and the referenced c2c phase three okay just to illustrate how things go in this case so if we include harmony harmonicity so believe phonos computer had a harmonic level we have as Morales predicted showed already we have a strong shift towards lower pressures of the enthalpy of the atomic phase but now if we include unharmonic effects using the zero point energy coming from ska we have the backwards effect so we go back in in pressure again so if we combine these together we get this final phase diagram and now we can you can see two main things the c2c is always the phase three stable but the the region here where the molecular phase cmsa 12 was very narrow in the previous phase diagram now it's much wider and and and the atomic phase is pushed towards a much higher pressures and we predict a transition from the molecular cmsa 12 to an atomic season four at about 550 gigapascal and now it's clear that the the experiment by Lou Bayer measure the transition between two molecular phases because now the atomic phase is clearly separated from the other two and our fan findings rationalize several experimental several experiments not only based on on conductivity but also optical spectra and reflectivity so we expect a shiny sample of course in the atomic in the atomic phase and the shiny sample is we should be expected above 550 gigapascal according to our error bars and that's uh in disagreement with the experiment by the assistive famous one who claimed the realization of the atomic phase of hydrogen already a 500 gigapascal so this is a medicated experiment and we are in disagreement with this now let's look at the eyes of the effect so thanks to the anharmonic treatment the quantum treatment of ions we can also study the the eyes of the effect so we found a strong isotope effect in both phase transitions both molecular molecular and molecular atomic and and unfortunately replacing hydrogen with the uterium doesn't improve the the situation because actually it worse it worsened the situation because atomic hydrogen is pushed at even larger pressures okay on the other hand we can check actually these the isotope effect measured by by Lou Bayer very recently in 2022 so he found an isotope effect at the transition between cms c2c and cmc 12 and that isotope effect of 35 gigapascal agrees very well with ours and as Lorenzo pointed out yesterday so these works are completely independent so we were not aware of Lou Bayer's work and we we've been happy to see that there was and this kind of agreement which somehow supports the the interpretation of this phase transition as two as a phase transition between two molecular phases okay so that's right hydrogen wrap up so I showed that that we've been able to combine quantum Monte Carlo with ska I showed that the importance of an harmonic contribution in to describe correctly the hydrogen phase diagram and so we predict a very high pressure for the atomic phase to be stable and we found a large isotope effects in both transitions and the the one by Lou Bayer associated with the direct gap closure is a molecular to molecular phase transition okay so now let's move forward how I'm doing with the weather with the time okay how long okay sick okay I try to accelerate you may need more so okay so the second part of the talk is about proton symmetrization symmetrization so proton symmetrization is is another process where you need a very accurate determination of internal energies and also an accurate description of quantumness for nuclei so it's shared by many by many compounds it's present in H3S the sulfur hydride we mentioned before in water ice as well for the ice seven ice 10 phase transition and but we'd like in this in this work we'd like to focus on a different system on a water cluster on a protonate water cluster and also in these kind of clusters you have the physics of proton symmetrization so why protonate water clusters are interesting because because the the additional proton the the charge effect is a sort of a probe to study proton hopping and hydrogen bond strength and you have usually in this cluster two limiting species you have a shared proton symmetrized proton in the zoom the species here and right hand side and you have a localized asymmetric if you like configuration where now you form an hydronium in the middle and with with a salvation shell given the given rise to the agon species so these two agon and zundal are known to be the limiting species of proton hopping mechanism in water and water clusters as well so we started building the zundal cation plus the salvation shell around around it such that we get a protonated water hexamer so we now we have six water molecules surrounding the charge effect the proton in the middle and so this protonate water hexamer is the smallest cluster including both limits the zundal like the symmetric one and the agon like the asymmetrical localized one in the same in the same cluster so if you plot this sort of phase diagram for the cluster you have the zundal like configurations when the size of the core is small at a short water water distance and at a large water water distance you have you are in the localized agon like situation so the bar if you are in the localized limit you have of course a barrier for the proton to hop to one molecule to the flanking molecule and the barrier is very small and that's a list of that's a list of values according to the methods for two distances plot the vertical line here sorry line so you see that the values in kelvin are close to room temperature so it means that not only quantum effects are important not only you need to have extremely accurate resolution of the energetics but also you need to include thermal effects okay and so in this work what we did we combined variational quantum Monte Carlo for the electronic part with a thermal bath provided by lancheven dynamics and we propagated the equation of motion for for nuclei using a path integral scheme in a path integral lancheven dynamics with an algorithm which is very efficient has been developed earlier by by some of us and it's called in this in this way but it's so it's a way to to integrate them this very efficiently and and we need these two to be able to perform these calculations are very challenging so we been able to run 40 picoseconds of dynamics using quantum Monte Carlo forces so if you will look at the core size here oops sorry we can we can now study the oxygen oxygen distribution function of the two of the of the core of the zoom the core of the cluster and we can now study the the thermal expansion of the core size as a function of temperature of course and we see that you see the results of both classical and quantum simulations and in the quantus case we have a very very small thermal expansion I mean almost negligible up to room temperature and then we have a shoulder here so so it means that quantum effects make the hydrogen bond robust in temperature that's the first observation which is very important and the second I don't know why it is sorry tilted so we studied the proton hopping through the distribution of instantons okay so we take an alternative approach we'd like to study the dynamics of the hopping of the proton by by setting the statistics of instanton events so the distant events are those events where the polymer describing the quantum proton in the middle is equally shared from by between by the left hand side and the right hand side of of the phase space is equally shared by the left hand molecule in the right hand molecule water molecule yes yes so now sorry for that I don't know why so so then we can do the study of the distribution we can compute the community distribution as a function of the distance and we saw we okay so that's let's see if it's okay so this is fine so by studying the community distribution of the instanton statistics we be able to find that there is a sweet spot for proton transfer around room temperature so in from 250 and 300 kelvin there is a sweet spot and and the sweet spot is explained by the the increase of short zoomed events so so this peak here is driven by if you resolve the peak in terms of contributions coming from different configurations it's an increase of of short zoomed events who's who's peak happens the same temperature range so it is this sort of surprising because short zoomed events are events which which is classically have a high energy because they are belonging to the repulsive part of the potential so we'd like to understand this why the short zoomed event are so important so relevant here despite being a high energy class in the classical framework and so we tried to basically derive a model a potential energy model involving the two relevant degrees of freedom the shuttling mode and oxygen oxygen vibration mode and by quantizing the the shuttling mode okay so this basically is an action dimensionality from the full quantum Monte Carlo driven md so by studying this potential and by quantizing the the shuttling mode we discovered that the zero point energy of the shuttling mode which is also tilted I don't know why so it's a PowerPoint version which is not the same as mine so you have you have a larger zero point energy on the egg inside and the lower on the on the on the short zoomed side and so this basically uh penalizes uh eigen light configurations and sort of favor any contrast uh zoomed light configurations and so let me wrap up about this water axomer so we found these importance of zero point energy to balance between eigen light and and and short zoomed light instances these explains uh because the potential now the effective potential now is our is harmonic is remarkably harmonic it is explained it explains the the the very low thermal conductivity found in the zoomed light core and also it suggests that indeed the short zoomed light configurations are relevant to these optimal spots sweet spot for for proton hopping due to the interplay between temperature and zero point energy and so let me just say the last word uh so the the combination of quantum Monte Carlo plus methods including quantum nuclear effects is at the infancy if I can say so because because we need if you like to improve efficiency to to take a larger systems and one of these one of the ways to do this is using machinery techniques to derive effective force field potentials we by keeping the quantum Monte Carlo accuracy and that's the work in progress and you see a lot of papers now uh exploiting one to Monte Carlo uh data sets to derive machine learning potentials and with that I thank you for your attention and uh I thank your questions thank you okay everyone thanks for the really nice talk and there's beautiful methods and we have just a very short time but please somebody ask a question yes there's one Sandro thank you Mikhail a very nice talk uh one of the additional interesting properties of hydrogen that actually nilashcroft pointed out in his paper is the fact that the uh vibrational energies have the same scale as the vibration as the electronic excitations you show this figure from experiments showing that the energy gap is 0.5 electron volts which is exactly the vibrational energy so of course the question is what about the Borno-Penheimer approximation right right yes indeed uh yeah that's another challenge of course everything here is is uh um within the Borno-Penheimer approximation in the study abatic uh so yes uh the solution to that is to write down you know a collective wave function including both nuclear and electronic use of freedom in a quantum in a quantum Monte Carlo framework at the wave function level for instance and and this will will include naturally also non-adiabatic effects between the two the two components so at the end you you're gonna aiming at writing down the two components with function okay so this includes an additional complication at the variation level maybe it's possible and I think there's I know there's some groups working on that so um but that's the way to go yeah to to to write down the two components with function okay so we have time for one more question um well I'll ask it is a sweet spot something that's really interesting about room temperature and the proton I so we know that proton transfer is you know um uh sort of important also for for everyday's life for the for the biological processes because salvation water salvation is is necessary for indifferent biological uh situations for protein folding so on so forth so the fact that the the efficiency of the proton transfer is maximized in a range close to room temperature is fascinating it's a fascinating perspective and I was surprised when I see when I saw this data and and so I think it's it's an interesting point I hope you're discovering something interesting so uh we must thank um Micheli again and the next paper here you are getting ready I'll pull out my get you all I hope please then correct me from the perimeter institute I don't have it it's 25 minutes plus five okay okay we'll we'll have this working soon here comes here comes help see hello can you hear me yeah sweet how does it work it doesn't work might work I can do okay okay go ahead please thank you very much um first of all I would like to thank the organizers for the opportunity to present my work I take the opportunity to thank ICTP as well I am the product of ICTP I feel like at home here I did my diploma program here I did my PhD between ICTP and CSUN so I'm like there I mean uh or not to be here one of my professors and I'm giving this talk so um as you can see in the slide I'm a research scientist at the perimeter institute I have affiliation at the University of Waterloo I'm also co-founder and city of my own startup so hard-wise in quantum support position of doing academic and uh entrepreneurship work I will tell you in a minute so I am lately very much interested in solving optimization problems but I'm interested in solving optimization problems using some of the techniques that you guys are developing here so quantum inspired techniques together with machine learning so optimization problems actually occurred in a wide range uh a wide variety of areas and we have the so-called traveling statesman problem which is the problem of finding the shortest path for a traveler going to a certain number of city ones you have a problem of interacting atoms that interact via some potential you want to basically find the configuration that minimizes that that system you have the problem of training your neural network nowadays people are training binary neural network because they want to save the energy that takes into training deep learning model you have protein falling problem of finding the native state of protein coming from an on folder state and you have portfolio optimization problem which is the problem of finding the best way of investing in the different assets that you have in your portfolio reach one of my students is actually working on with quantum protocol and you have many more problems so why are all of these problems interesting for me as a physicist is because they could be cast in the form that I understand which is the form of an Hamiltonian basically and solving them with whatever constraints that your problem has is equivalent in finding simply minimizing that that Hamiltonian or finding the ground state if you want right so we can typically use all of the methods that we've developed over the decades to basically find ground properties of classical quantum system to be able to solve all of these optimization problems but there's a problem some of them are very hard to solve like for example the disorienting glass in the random random logical field find the ground state among two to the n possible states and right so typically what people do depending on the application that you're interested in you kind of relax your initial absorption of wanting to find the lowest configuration near optimal solutions are okay so basically people develop heuristics method that find your optimal solutions one of the most popular method so another thing that I like to see is like visualizing my optimization premise some sort of very crazy landscape that has a lot of local minima a lot of solder points and then solving the optimization premise finding the deepest valley in that landscape or finding a near optimal valley basically so one very popular method is basically exploring that landscape using thermally or fictitious thermally activated processes right and this has been inspired by a method in metallurgy whereby you basically to make a material more durable you kind of heat it up at a very high temperature so that you can explore high energy state and you slowly cool it down so that the atoms we arrange them serve in configuration that minimizes the free energy right and so this is the so-called simulated annealing method that was invented in IBM Watson to basically optimize chip design another project the game is basically to instead of using thermally activated processes you can use quantum tunneling to basically explore your landscape and so you have the so-called quantum annealing method which aims at yeah finding the ground state or the global minimum using quantum tunneling and that has been also implemented on some dedicated hardware here I'm mentioning two of them because they are actually available on the cloud and you can simulate quantum annealing there all right so basically this is the practicing of either searching for the ground state either using thermally activated process or quantum activated process which one is better but we don't know I mean usually it's like you risk is kind of trial and error but in a lot of applications people have found quantum search to be better than a classical search but what I would like to highlight is as most of the techniques that have been used either to benchmark the quantumness of some of the devices that we are building or to see whether there's any kind of quantum advantage in using them is by actually simulating the dynamics of either simulated annealing or quantum annealing on my laptop or whatever HPC architecture all right and what people have been doing I've been using to do that is using Monte Carlo methods either classical Monte Carlo methods for this emulation of simulated annealing or using quantum Monte Carlo methods for the emulation of quantum annealing so probably you recognize a lot of those methods here and those methods were not designed to solve optimization problems right they were designed to basically find like the previous top equilibrium properties of either classical or quantum system but here they were repurposed in order to solve optimization problem so we're inspired by that and by very recent work that we're actually using neural network to simulate equilibrium properties of either classical system or quantum system and then we ask ourselves whether we could repurpose them as well to basically solve optimization problem in annealing paradigm so next I will give a very brief introduction to variational Monte Carlo which is one of the bedrock methods that we use to emulate annealing and VMC is basically the quantum Monte Carlo method that is used to find a once a properties of quantum system at zero temperature it does that by simply minimizing the so-called variational energy expectation value of your quantum boundary and over your variational state so by definition your variational energy is an upper bound to the true branch of the energy so you have you just have to minimize it to optimize the parameters so I would like to point out to the talk by yesterday because in principle that and that would be whatever yesterday we had a quantum secret whereby basically you do variational quantum eigen solver to be able to represent the quantum state with a quantum secret here we use actual networks and then we have a typical thing you will present you replace quantum mechanical expectation value with statistical ones and then you optimize and that just taking the gradient and taking whatever flavor of the optimizer stochastic gradient descent mentum stochastic reconfiguration here I would like also to give a tribute to Sandra who is like was you saw in the previous talk he was really a leader in basically designing uh and that's for variational Monte Carlo especially stochastic reconfiguration techniques that nowadays is like state of art as well to basically optimize neural network all right so next um which answer do we choose so we actually choose neural language models and I will tell you in a minute in a minute why neural language models are very powerful probabilistic models that I used to find correlation between words in the language so basically when I learned that correlation is able to predict the next words giving previous work and I believe that all of us are using that when we write emails nowadays right so I'm going to give you a very like vanilla idea of height works with uh one uh towards a regressive model called recurrent networks all right so imagine you're writing your email and then you put the input of people you're writing to with the subject so basically what the neural network does is that it predicts the next work it has learned the conditional probability distribution of previous in the next word it takes that word that it has sampled to predict the next word and so on and so forth so you can immediately see that if you take the product of those conditional probability distribution you've learned the jump probability distribution of sampling the first word the first words in your sentence giving the input but why am I interested I mean you are physicists why do we say sample words so the point is that this kind of autoregressive sampling could be applied to instead sample a spin configuration or a qubit right and you can literally sample a qubit or a spin at a time using the same autoregressive process and why is this interesting maybe I give a little bit more detail so basically you use the chain word of probability distribution to be able to sample or to learn the jump probability distribution of sampling a spin configuration and what we are actually learning from the RNN here is the condition is the conditional probability distribution that we parametrize with some parameters lambdas that are literally the weights and the biases that lives inside your RNN set right and then once you have the probability distribution we use a soft called softmax layer indicates that make the probability distribution by construction normalized and that's going to be pretty important in in couple of slides and we can represent the wave function just taking the square root of the probability distribution okay so one of the reasons why we use autoregressive neural networks there are other neural networks that have been used things like respect both my machine convolutional networks but the point is that if you use dotnet's work to basically represent economic body wave function you need to figure out a way to sample right because here we use an approach that is similar to reinforcement learning we don't have a data set that we are training on use you generate the data that you train right and imagine that you are basically trying to minimize a spin glass and you want to to generate samples of Monte Carlo auto correlation times are going to kill you right so we want to have a smarter way than just building Markov chains to sample here we don't have any Markov chains so we have no auto correlation times practically speaking is directly palatable and yeah it's normalized to unity but as a catch you can tell me that quantum mechanics is quantum wide rough square root of probabilities well you can also represent complex wave function with that and we can simply parameterize the phase let's say of your your your wave function using basically still your RNN but putting it through another layer all right and you do the same thing minimize the the variational energy the variational energy so now we tell you about the RNN architectures that we use um in most of the local amiltonians that we are looking like we use the tense tense wise RNN cell and we saw that compared to a vinyl RNN cell where basically you have your so-called hidden state that encodes the correlations uh that learns the correlation between the spins in in your lattice it interacts with the local Hebrew space just through your activation function but we chose a tense wise version because it was a month more compact representation and when we we run simulations on the quantum mixing chain like criticality we look at the variance of the the variational energy with respect to the training step you can directly see that the tense wise RNN converges faster at the lower accuracy so just as a reminder variational Monte Carlo has this very nice zero bias property so I mean the lowest it is the more accurate your answer is to the ground state all right and another thing that we noticed is that when we had this order in the system instead of having the same what we call weight sharing that is at every lattice side we use exactly the same weights and biases of the RNN when we have different weights and biases which what we call not weight sharing there's a dramatic improvement or a dramatic difference in the ability to represent basically the the wave function and here you see that basically it's almost like toward of magnitude more accuracy when we use non-weight sharing so somehow the RNN has to see on their lying macroscopic details of your Newtonian to be able to represent it well and when we look at other using glasses the so-called shiitoken path which is an all-to-all interaction model with random interactions and we we found that using a different kind of RNN architecture the so-called directed RNN cell that uses this the so-called skip connection we saw one yesterday and it was introduced skip connections were introduced in machine learning to deal with the problem of vanishing gradients here we saw that it was pretty efficient to be able to capture the long range correlation that we have in in the shiitoken path and again when you see the variance of the variational free energy which I will define in a minute you see that you can have up to one octave of magnitude of accuracy using this kind of skip connections so one thing I would like to highlight also because what we did in the previous RNN cells were like more of trial and error and usually we don't want that but the smart way of going usually is implementing symmetries in the RNN architecture so here in the first panel on the top left I'm showing some results obtained by some people in my group where they basically simulate the xy model in 1D and they showed the RNN when they encoded symmetry the u1 symmetry and when they didn't encode the u1 symmetry and we saw that the symmetry converged faster and there's another results on the g1j2 model in 1D where they showed that when they encoded the martial sign in the wave function they basically have orders of magnitude more accuracy so encoding symmetry is important and basically on the other panel what they showed is basically when they plot on random direction the lost landscape where the training is is obtained in this case is basically the negative like like so the left data they actually do on supervised learning where they train from data they obtain probably from DMRG or Conor Monte Carlo and they show that basically when you don't have symmetry it takes time for you to find the global minimum but when you have symmetry it's directly converged very very fast so yes so we know that RNN was an universal approximators but if you don't encode symmetry it takes more or less forever it's not so accurate when you want to train it so it's a good thing to implement symmetry so another thing I would like to mention since my talk is not too much of electroinstruction I want to make sure this very nice review that was put on iCAD recently where they discussed about different network architectures for fermions basically both in the first quantized regime and in the second quantized regime so please have a look maybe is the only thing you take from my talk all right so next I'm going to talk about variational quantum annealing how many minutes left tell me it's good sweet all right so I motivated my talk by saying that you could basically solve an optimization prime wave maybe a quantum annealer by using quantum fluctuations by using quadruple possession as a search of basically this kind of configuration landscape but you could view it in a different way you could view it in the spirit of adiabatic quantum computation so the way we actually view it in a variational quantum annealing is that when we have a RNA typically you initialize it with random weights and biases right so when you initialize random weight on biases so quantum annealing you could see it as an interpolating method right I have an amyltonian that is easy to prepare I can very I can easily find a ground state but I want the amyl the ground serve an amyltonian that I don't it's hard to prepare so I can slowly or adiabatically interpret between an amyltonian that is easy to prepare to the amyltonian that is hard to prepare that's kind of the idea of adiabatic quantum annealing and you do need to do it slow so we kind of use the same ideas here there's this amyltonian where all the spins point in the x direction that's easy to prepare the way we do it we simply do brain descent steps and we land on the the energy of that amyltonian that is easy to prepare next we change the amyltonian we introduce basically a problem amyltonian that is in glass and what happens is that basically we shoot out of the instantaneous ground state and we need to prevent some you need to perform some gradient descent steps to fall back on it we reduce again some quantum fluctuations we shoot out of the instantaneous ground state we perform gradient descent steps to fall back on it and so on and so forth by the time you have completely reduced all the current fluctuations in your system hopefully we are we should find ourselves in the ground state of raising glass in that case so we came up with a theorem that we do a variational adiabatic theorem that tells us the number of brain descent steps we need to perform to remain adiabatic and its scales is bounded with the minimum gap and the minimum gap square that you have during the annealing and somehow in that sense we still have the complexity of say you pass through first order phase transition exponential scaling up you still have an exponential kind of scaling so there's literally no free launch so recap we use variational Monte Carlo with a time dependent amyltonian to be able to emulate quantum annealing your classical formulation is pretty easy you just have a variational free energy that we also minimize because it's also an upper bound to the true free energy right and for the variational classical annealing you also use a time dependent schedule and one thing that is also important is that you need to be able to represent exactly von Neumann entropy so if I have an IBM that is not normalized or it's a computational network that is not normalized I cannot basically represent the entropy right because yeah I simply cannot whereas when you have a probabilistic model that is normalized is very easy and is computational efficient to to represent or to compute the von Neumann entropy so that will present the results so here we have variational annealing on randomism chain we have an amyltonian we've spins that interact with different kind of random interaction on the chain we know what the ground set is so we monitor the residual energy the difference between the amyltonian we obtain at the end of the annealing and the exact one we try different system size something I forgot to mention is that the difference that we found between an amyltonian base approach where we minimize the amyltonian or non-supervised learning base approach where you need to have your training on data usually when you are training on data using a negative look like you need a lot of data to be able to learn the probability distribution but for us we don't need a lot of data we usually lose like 50 samples to train which is very impressive but once we've trained we can generate an average train number of samples so the end of training we generate a million samples basically and we can do it very very quickly very fast and we do we do averages over also 25 random realizations of the disorder so here we see that basically as we increase the annealing steps which means that we become more adiabatic than doing some sort of quenching indeed the residual energy is going down something that was so was surprising to us is that actually here the classical emulation of annealing with neural networks is supported the quantum emulation analysis and the previous results showing the contrary that is emulating annealing with metropolis Monte Carlo with the quantum case was actually superior so we have kind of a reverse here with this kind of variation and we suspect that maybe is the use of the explicit use of the entropic term that that brings that advantage and another thing that is also interesting in that in that paper they actually did a time evolution of quantum annealing of what the current quantum annealer should have and on this one of them is in chain it has a logarithmic scaling whereas we have a polynomial scaling so which means that algorithm for this system size I need to repeat the current computer which was pretty cool to see and then we increase the complexity we went to the two-dimensional evidence and there's some spin glass and again we had to change the neural network and neural network the autoregressive sampling takes into account basically the locality of of the 2d is in glass here so we do this kind of sampling in 2d so one thing that is interesting again is we saw that vc was superior to vqa but another thing that makes us we wanted to check that actually annealing is is is helping the simulation so what we did we directly optimize the problem we directly optimize the problem and without any annealing that's the red data point with respect to the number of a billion descent steps we see that with large large training steps we basically lower the residual energy but it's orders of magnitude higher than the rational quantum annealing and rational classical annealing which means that somehow having an idea of annealing in the simulation is helpful right so based on that we know we knew that there were previous results by by uh Santor and collaborators on the same model but doing an emulation of annealing using uh Markov chain Monte Carlo and we did the compression and it was amazing so we saw that for long annealing time we could have up to three others magnitude more accuracy using a rational classical annealing compared to both simulated annealing and simulated quantum annealing implemented particular Monte Carlo right and this was on 1600 spin chain we try also a straight on key pattern model here we saw also some advantage it was not so drastic and we saw that most of the time the the Markov chain base annealing techniques got stuck in local minima and we also look at the probability of success of the different instances and vna or vqa in this case was almost odd one when you find the solution is almost with perfect accuracy we even look at uh basically we did uh pca on the the the solutions that we the configuration we obtained at the end of the annealing and it was cool to see that we could live in sample in digital advanced state here in in sq so next we wanted to tackle another problem like there was this paper discussing the fact that as i mentioned there is basically a time scaling in in being able to solve an optimization with quantum annealing and that's the the gap the inverse of the gap square right and for for primes where basically you pass through a first order phase transition you have an exponential scaling up which means you need an exponential amount of time but then there was this very nice paper by Inish Murray and this collaborator showing that actually you could enlarge the phase space and by enlarging the phase space you could find path that avoids the first order phase transition and you pass through a second order phase transition so we have a polynomial scaling up and because of that you will have an exponential enhancement in solving quantum annealing so here was the absolute of his of his talk and he said he said that was the definition of non-sequesticity by the way um in non-sequestic amyltonia is an amyltonia and in which you have your off diagonalize terms that could be positive and negative so it makes common Monte Carlo traditional common Monte Carlo to be not simulatable because of the sign problem the violence of the sign was exponential racism size but VMS says it was not intrinsically as a sign problem as I mentioned so we asked ourselves whether we could simulate this version I mean version upon algorithms and whether we could actually see the advantage by passing through a second order phase transition so we look at this pspin model um the HP which is a basically a minfin model with range of interaction p and the range of in interaction dictates somehow um the phase transition right the red lines here are lines where your first order phase transition and the blue lines are the lines with second order phase transition all right and we look for example as as proof of principle for p equals to five so if we decide to have an annealing schedule where we pass through the second order phase transition we just check that indeed we could simulate the dynamics right with rational quantum annealing we pass to the first order phase transition we can also do the same thing here next we scale the system and we look at the success probability when we take a route passing through the first order phase transition and when we take a route passing through the second order phase transition and we see that there's there starts to be a shift as we increase the numbers of of spins in our systems and when we look at basically the number of annealing steps that we need to have a 99 percent success probability we see that it's way much more favorable passing to the second order phase transition compared to passing to the first order phase transition one thing that was quite surprising for us is that we didn't see any exponential would have expected to see number of getting the same step to be exponential in the or the normal annealing set to be exponential in the system size but maybe it's because the system size are not large enough but at least we can see that the route by which you pass indeed has an effect so we check another system this is called new man more model okay I will go through very fast in my model is a model that basically does have an explicit development of a speed is exhibit glassy dynamics as signature of first order phase transition there are fractions excitation that makes it impossible to simulate with low temperature with Monte Carlo we try variational quantum annealing and variational classical annealing and we saw that basically annealing dynamics was totally unstable and yeah so it was totally chaotic and we're asking us of why is it chaotic we brought again a method from machine learning where they basically try to understand why a network without skip connection was worse than the one with skip connection by visually in a loss landscape of of the prime and we basically did the same we visualize the loss landscape during variational classical annealing and we saw that indeed you are the characteristic minimum in the beginning of annealing which is what we are supposed to be but then it becomes totally chaotic at the end so that's the reason why is it's not efficient for that so there's basically no free launch just looks to be for some problem there's a conservation of computational complexity and with this uh I these are my conclusions annealing search in the variational landscape seems to have performed the search in the configuration and landscape we saw VCMO efficient and VQA and S and SQA but we said that we advocated a good candidate to solve real world optimization problems VQA is also a good candidate to benchmark front manila with non-stochastic and maintenance and we should not just look at representative or expressivity when we use neural network ANZAT for variational Monte Carlo but also we look trainability issues and visualizing the last skip may help us to design better ANZAT with that I thank you thank my collaborators and I'll be happy to take any questions thank you here we have some questions questions yes maybe I missed the point many of these Hamiltonian are fully degenerate in the ground state so for example when you do from the adiabatic when you were doing the adiabatic connection the the Hamiltonian with only the spin interaction can have for example if the j is uniform this can have whatever solution you so there is a way to select a specific ground state or how you solve the the potential fact that your Hamiltonian has many ground states in your search yes so basically there's a if your machine learning model is able to capture the multi-modal distribution of your ground states then in principle with autoregressive sampling you can sample all of them right so there's there's a part that I skip very quickly for a human model model where basically I oops I showed right here we have four degenerate ground states and at the end of the annealing when I generate let's say 100,000 samples and I basically plot an histogram and I see that I find all of the foreground states this is when I don't have more collapse but if I have more collapse I probably collapsed one of the ground states and maybe if you have a penalizing term in the in the loss function you might bring a bias to one ground state then I don't know for example you can break the z to symmetry by hiding a longitudinal field to make sure that you go in one ground state of your normal yes very nice talk thank you and I'm curious about have you tried the real model I see you have some more only include the Hansenberg interactions have you tried to like a real model like include a very long range interactions like Hansenberg interactions you calculated more than 20 nearest neighbors like included the direction scheme real interactions and other interactions so you will have a very complex system have you tried this I mean I should resolve on Sheridan Patrick is all right I'm sorry so the shake on the Sheridan keep Patrick model yes I mean your your quantum annealing models to find the ground state your what do you use a spin glass I know it is very famous model but have you tried to use a DFT calculated model a real system a real material for example you have you have a system that you have your Hansenberg interactions you calculate to like a more than 20 nearest neighbor and you also include direction scheme real interactions and it will be a very huge interaction matrix have you tried this with your method is that efficient okay so if I understand you are asking whether I tried a model that has many interacting of nearest neighbor interactions I did I mean I believe like the shredding tongue keep Patrick talks to all of his other neighbors so in that sense it's kind of infinite dimensional somehow yeah so basically these are the results here for the Sheridan keep Patrick model right we can talk later maybe I don't understand the question okay okay anything else any other questions not over there well I'm told by the organizers that we really have to be very careful now to to start the next session on time right so we'll we'll take we'll oh first we thank our speaker again thank you back here at 330 so