 is there I see people are still joining so maybe yeah we can wait a few minutes the streaming is on I make a pun the streaming is is on yes so maybe yeah so do you think we can start yes maybe we should we should okay okay so hello everybody it is hello everybody it is a great pleasure to introduce professor francesco maury from university di roma la sapienza professor maury is an excellent scientist with an outstanding cv according to google scholar he received over 63 000 citations and holds a hirish index of 89 francesco maury graduated in 94 at university de genova and seaside resta then he moved to university of california at berkeley and returned back to europe in 97 and since then held a number of positions at various institutions including university de genova university pia america cnrs and today he's a full professor at university di roma la sapienza he published about 270 papers among them 40 physical letters nine nature and one science paper he authored or co-authored a number of very influential papers including quantum espresso papers and a paper on graphene entitled raman spectrum of graphene and graphene layers that was cited 40 000 times a few others of his research papers also received on the order of 1000 citations each professor maury acted as a p.i. of several european and national projects his research activities are focused on the prediction of the physical properties of complex materials using first principles electronic structure methods and the development of original conceptual theories methods and algorithms to treat interacting electrons in particular he is active in research fields such as computational spectroscopy phonon mediated superconductivity carbon nanotubes and graphene unharmonicity phonon-phonon interaction charge density waves and thermal conduction amorphous materials mineral physics and geochemistry today he will present us a special guest lecture entitled phonons and second-order structural phase transitions in unharmonic quantum crystals professor maury the floor is yours okay thank you so let me share okay you see the screen yes okay so thanks a lot for invitation at this school and the the topic i will speak about today is about the our recent development in in the harmonic circuit harmonic theory which is a old approximation we date back to the 70 but it had never been used in conjunction with density function first principle calculation and the idea already is to use this theory to treat cases where phonon interact among themselves with a strong and harmonic coupling so the the topic is pretty broad and we are since we are working for many years and so i will not have time to cover all the application either all the theory aspect and so i will focus in this talk particularly on the study of second-order phase transition that involves phonons and the associated phonon softening and so without first of all the the work is a collaboration involving many people uh the the most relevant people are outlined here and the work was started with yone rea matthio calandra and myself in paris and as i continue with the contribution raffaello bianco contributed a lot in the aspect of the study of second-order phase transition in Lorenzo monacelli who also was the keeper son to make his approach a practical call for to treat a very large system and also thanks to him that now we have a code that we can distribute because his interface with quantum espresso will describe this in the end of my talk so this is the outline of the talk i will start uh discussing the the second-order phase transition that involve phonons and the phonon softening phenomena that occurs in in close to the phase transition then i will present our approach which is to develop an approximated uh expression for the free energy that can deal with such kind of phase transition and the our approach is based on the second-syntheremonic approximation so we'll discuss what it is what is the second-syntheremonic approximation and then i will show you what is possible not only to compute the free energy but also the first and second derivative of the free energy we expect to the uh ionic position and these give rise to the possibility to describe within a landau theory type of approach a phase transition driven by thermal or quantum situation then i will show you an application a practical application to two cases presenting charge density wave which is uh charge density wave is a specific second-order phase transition and i will show you oops the case of neobium selenium 2 and neobium sulfur 2 which are two decalcogeneite that can be also exfoliated to have a single layer a single monolayer but and that present charge density wave both in the bulk of the monolayer phase finally after the application i will specify a little bit more in detail our practice we perform the calculation and the key ingredient is a stochastic approach similar to a Monte Carlo type of approach to solve the equation that enters the second-syntheremonic approximation and the evaluation of the cool virtual of the free energy and this concludes my talk now so let's start the uh the so what is the harmonic approximation so in the harmonic approximation we consider the motion of nuclei uh considering the electrons adiabatically uh integrating out of the electron the electron is of freedom so we do a bonopenheimer approximation so for each ionic position we can minimize the electronic energy and obtain a bonopenheimer potential energy surface for nuclei and so once we do that we can find the minimum of or the absolute minimum of this potential energy surface and develop the the potential quadrat quadratically around the minimum and this define small oscillation around the minimum so if we do so we obtain a harmonic vibration which is in in a crystal are characterized by equals more into mu in a mod index which indicates in uh within this approximation so this is in this framework what we have is that the uh if we prepare the system in uh displacing the atom from equilibrium then we start oscillating and we oscillate with no damping because uh each phonon uh constitute a decouple uh Hamiltonian with the other phonon and so uh the vibration lasts forever which is uh does not correspond to uh reality because indeed phonon has a finite lifetime and one manifestation of that is that uh the for example the thermal heat is not transferred baristically from one side to the other of a crystal and this is because of the phonon interact with the other phonon and their environment and uh acquire a finite lifetime so uh what are the origins of lifetime there are in a string sick phenomena related to defects but also even if you have a perfect crystal there are interesting phenomena to do the fact that the phonon interact with each other and uh they interact through the expansion of the potential beyond the second order and uh and uh and uh in in a if we can expand using a theory theory Taylor series the potential beyond the the second order we obtain uh a uh the expansion related to the third derivative of the potential especially on this position we obtain uh the fact that one phonon interact with other two phonon and this vertex we conserve energy and uh in quasi momentum and in the higher order is the first phonon vertex the five phonon vertex so for us all now if the the the motion around equilibrium is small so in presence of low temperature of in presence of large masses where the zero point motion can be considered to be small we can remain uh with a perturbative description keeping the lowest order of the theory and uh if wanted to uh consider the the lowest of the correction to the anharmonic uh phonon theory we have to consider the three phonon vertex and the four phonon vertex actually the two give rise to diagrams which are of the same order in terms of the displacement of nuclei from equilibrium and this is a case where we can treat the weak anharmonicity and we can use this diagram for example to compute in a very predictive way the phonon lifetime and for example also to compute for example thermal conductivity properties okay this is correct if we can use this uh this expansion but there are cases where the anharmonicity is so strong that a perturbative expansion does not uh converge and this is the case where the fluctuation around uh the equilibrium position is very high so two examples of uh our system where the zero point motion is very large for example system containing hydrogen so if you want to do the phase diagram of pure hydrogen for example under pressure or of hydrogen based uh temperature or close to the temperature superconductors in this case the the zero point energy situation of hydrogen are huge and so an expansion like the one uh described here is not appropriate another example is uh if we increase the temperature the temperature a lot and so if we reach temperature closer to the melting temperature also in this case the situation are very large again an expansion uh based on a Taylor expansion could be inappropriate the last case which is the one i will focus in this talk is uh what's happening uh close to a dynamical instability so near a second order phase transition like for example a ferroelectric phase transition or cernesity wave transition actually the two have a very similar uh uh nature from the point of view of phonons uh with the inference between the two is that the cernesity wave usually occurs in metallic system and ferroelectric instead of transition occur in insulating systems but in both cases what we have is that we have a very symmetric uh phase at very high temperature where the situation is at large and when we cool down the system what we have is that the system and give rise to a phase transition to a lower symmetry phase and in the high symmetry phase the the harmonic phonon usually are uh imaginary this means that if you compute the cool pressure the cool pressure of the energy in the high symmetry high temperature phase usually this cool pressure is negative so in this case we cannot uh if it's negative we cannot do an expansion around an harmonic solution because the harmonic solution doesn't exist okay so this uh gives you the motivation of the context in which we are moving so uh i want uh you to give an example of cernesity wave is the one that i will study later and uh so let's consider uh niobium c2 which is a a layer compound uh in which uh each layer is constituted by a a this is a side view a unit where we have uh a transition metal uh in the in the center in this case niobium surrounded uh by two layers of a a calcule gaming atom so sulphur selenium uh or tellurium and uh niobium this lanite is at least the first system known is of dimensional higher than one in which has been identified the occurrence of cernesity wave uh we can have a single layer but we can also have a a crystal constituted by many layers where uh the different layer valentine valence bonded so it is a weak bond in between layers and uh what's happening is that we have a high symmetry phase uh that is stable above 33 Kelvin and if we cool down the system below 33 Kelvin we have a a symmetry breaking in particular in this case the breaking is of the translation of symmetry and indeed for t lower than tc we observe a superstructure with a periodicity uh close to the three by three uh by one so so we need to take the unit cell and multiply by three in the two dimension of the plane and instead is one in the the perpendicular direction and and this can be seen by for example diffraction experiments now if we look at the funnel uh this is what's happening to to the funnel uh dispersion and uh these are experimental funnel dispersion measured by x-ray and elastic scattering on a on a single crystal of bulk and niobium selenium 2 and uh each different symbol corresponded to a different temperature so the black symbol for example are high temperature measurement and these are shown the funnel dispersion of of a funnel branch the one that get driver with the phase transition as a function of of the funnel momentum as you can see that if you are at high temperature the funnel has a ferrite energy now if we lower from 250 to 50 which is the right the funnel gets soft and uh at the phase the transition temperature of 35 Kelvin we get the the the blue uh triangle and so the funnel touches the zero energy axis this means that it costs zero energy to distort the structure and the at this this uh momentum corresponds uh closely to the one necessary uh to to to every by three by one uh reconstruction now if we lower again the temperature the system undergoes a phase transition and the funnel here in uh became again uh positive so goes up now we can take this funnel momentum and study what's happening to the funnel frequency versus temperature and uh these are being experimental data you can see that we have a decrease of the funnel frequency and the funnel frequency goes like a square root of uh t minus critical t uh to zero at 43 Kelvin and now if we would continue uh below it goes back up okay so you can see that uh the uh phase transition occurring in diffraction uh is uh if once originated by a softening of this funnel and that if one looks to the pattern of this funnel indeed correspond to the the pattern that is found uh for t lower to see okay this is the experimental data now the idea so is to have a theory able to cope with this phenomenon now uh so our goal is the description of phonon in strongly an harmonic regime using a principal approach and to this uh goal what we use we we we use a the second system harmonic approximation which was introduced by uton by catholic 55 so it's a very old theory and uh the nice thing is that uh even in in a strong non perturbative regime for example interdensity wave or a ferroletrics which give rise to a similar physics uh but with a phonon at some center instead a phonon at a moment a quasi momentum different from zero uh if you consider this strongly non perturbative regime uh if you do an experiment like the one that i showed to you and the the the experiment a major uh object which looks like phonons even if we know that it cannot be uh harmonic phonons because an harmonic phonons by itself is something which is temperature independent and so phonon can still exist and also can be measured if it's justified to develop a theory where a what uh a dresced uh phonon dresced quasi particle that looks like phonon can describe the system now what is the harmonic approximation the sequence is the harmonic approximation is the theory which has the same philosophy of archifoch the archifoch approximation was developed for elections so for fermions the circuit is harmonic is a similar theory that applied to boson basically which are phonon and so in this theory we we use like in uh in archifoch we use effective non-interacting electrons that minimize the total energy uh using the exact electron interaction this is the archifoch uh philosophy and the psychosis harmonic approximation philosophy is to use an effective harmonic non-interactive phonon like in archifoch we consider non-interacting electrons that minimize in this case not the energy because we need to have a temperature effect so that minimize the free energy uh considering the exact an harmonic to all order potential so we don't do any approximation to the phonon phonon interaction but we approximate the solution considering uh a solution generated by an effective non-interacting harmonica metronia no so this is the philosophy let's look how this will honestly translate into equations so to uh to this goal let's consider the free energy of uh uh nuclei so we need to define so in this case the electron does not exist so we consider nuclei like one two particle moving in a potential given by the electron which is uh will be this v of r and and what we want to consider is the uh free energy which is constituted by the internal energy minus the temperature time the entropy and in an exact way we can obtain the uh the free energy minimizing the the trace of uh the product of the quantum Hamiltonian for nuclei uh times the uh nuclear density matrix this is the u part then we have a minus t s the minus t s can be expressed like the Boltzmann constant the temperature and the trace of the um of the density matrix times the logarithm of the density matrix itself where rho is the embodied correlated density matrix and h is the nuclear Hamiltonian where uh which is constituted by the kinetic energy of all nuclei and the potential energy of all nuclei so this r is a vector which contains free and uh elements which are the free and Cartesian coordinate of all n uh and nuclei or an atom considered in our system now this is an exact solution and what we should do to get the free energy we should minimize uh this expression in respect to all possible density matrix that has chase equal to one so if we do this minimization we obtain the uh the exact free energy that this minimization has a formal solution and the formal solution in this one so uh the formal solution is that rho is the exponential of h where h is an operator so rho also is an operator the quantum numerator is divided by kt uh and z minus one is just the normalization so it's the trace of the exponential of h divided by kt now the uh this expression is exact and the the problem is that it is extremely complicated to compute to compute because h contains this potential this potential is embodied potential and so to uh to solve explicitly uh to obtain this uh exponential uh uh operator and imply resolving a many body problem which is uh uh exponentially complex respect to the number of okay so we should so this is exactly the result so let's look at what is the the approximate uh how we can obtain an approximate solution and in this case we have we can define a functional which depends on a trial Hamiltonian h so I put a tilde here because it is a is a is a trial free energy which depends on a trial um Hamiltonian and and uh in the philosophy of uh Hartree-Fock uh we I replace this exact expression with a trace of the exact Hamiltonian so containing the exact uh interacting potential but here I replace the density matrix by an approximated density matrix which is the density matrix which corresponds to this trial Hamiltonian now uh what I have is that since the the the exact free energy is obtained minimizing respect to all possible density matrix uh if I use an arbitrary Hamiltonian different from the exact one here I get a free energy which is higher than the exact free energy and the quality holds only if the trial Hamiltonian coincides with the exact Hamiltonian so what I have is that we have a variational principle uh constructed on the trial Hamiltonian so far I just did a change of variable so I do a minimization spectrum if you want h tilde instead of doing a minimization state to to to know but I didn't change I didn't show you any attenuation yet because if I do this I can get the exact solution now uh the uh you can use this variational principle to define the psychosis of the harmonic and in this case instead of having an arbitrary trial Hamiltonian I restrict uh ourselves to a an harmonic Hamiltonian which in the most general uh case can be expressed in this way so we have the standard kinetic energy and replace the potential with an harmonic potential and the harmonic potential has two different kinds of parameters which are the uh RC which we call centroid we'll see later why I said called centroid which is the the position uh around which we expand we have our quadratic is the zero of our quadratic potential and then a matrix D which represents the curvature of our harmonic harmonic trial potential now uh this is the most possible general uh Hamiltonian again I remember that this is not a single atom but they represent all and the calling of all an atom so this is a is a freeing by freeing matrix and this is a freeing vectors and so these two objects this freeing matrix freeing by freeing matrix this freeing matrix represent the degrees of freedom they describe the most generic trial harmonic Hamiltonian now uh the shot this is the American approximation consists in finding the best uh RC and D that minimize these free energy and these correspond so once we do this minimization state to these two parameters we obtain the set position harmonic free energy okay now uh so basically the free energy has been applied in development in the 50 and applying the in the 60 and 70 to describe the uh crystal of noble gases where the ion potential that was described with uh an analytic form and so uh these minimization was done either fully analytically or numerically with a little effort and so our goal was to extend this approach to deal with any possible system uh the total energy is evaluated using uh a first principle approach like this is fashionable theory or even for example quantum Monte Carlo or K. K. Rastore method which are made or which are even more precise in this fashion and theory and in this case the total energy and forces uh usually this could provide the total energy and often forces for free once we complete the total energy but the only uh point is that this uh total energy and force evaluation are expensive so what you want to do is want to solve this harmonic approximation limiting the number of calls to a a nabination engine which could be quantum espresso in the philosophy of the present school and so we want to minimize the call of quantum quantum espresso to evaluate the total energy forces and so to this goal we implemented a Monte Carlo like uh uh so a stochastic approach to the solution that considers the harmonic approximation uh to minimize the number of calls to be a initial total energy and force engine so this has been given the methods in these free papers where in the last one we also show how to compute not only the to minimize the internal position and to obtain uh so uh RC and D but also to optimize the itself of the crystal so to do a full minimization both of of of uh lattice parameter and internal coordinate of our system okay uh now the question i would like uh to rise is that uh i would like to discuss the meaning of the auxiliary harmonic Hamiltonian so at the end of our minimization what we obtain we obtain this free energy but as a side result we obtain also uh this matrix D and the position RC that correspond to the minimum free energy so the question is that so we have an harmonic Hamiltonian and the question is that if this harmonic Hamiltonian has a physical meaning or just an auxiliary uh object now uh let's go back to the electronic type of calculation uh if you consider density fashion theory for example we know that in density fashion theory the goal of the theory is to obtain the ground state energy and its derivative and the cone shamb orbitals and eigenvalues are auxiliary variable but it doesn't have uh necessary a physical meaning apart for them and apart peculiar uh situation so in the same manner this each field is an auxiliary Hamiltonian so in principle it's not related to an a physical observable could be just an auxiliary field so so the question is that uh if there is a connection between the these h tilde and for example the phonon that has been measured for example in the example i gave to you in uh in the in our chart basically wave system or in the ferroelectric system for example it's so the question is like if you have direct access to following stability you look into these uh matrix d and rc so if you have access to this situation okay so so so the question is like what are the meaning of these two variable rc and d t so so in this case uh we can evaluate uh so this is again just repeat our auxiliary matrix and uh from the auxiliary matrix we have the density matrix and the density matrix which is uh rhod hilda is the one obtained with this h tilde and given an observable with just the pain on the anionic position we can just consider the trace between this observable and our trial uh or optimal uh harmonic density matrix and since uh if you just depend on position what we can translate this trace in the integral over space of the observable where here is operator here is is a classical variable so we do the integral all possible position uh here is this very possible observable which depend on position time and probability distribution probability distribution depends on rc and d and is nothing but the diagonal part of the density matrix and the uh the nice thing is that if you consider an harmonica mitonia this uh this probability distribution can be computed analytically so is a reality function that i'm showing here just for sake of uh clarity but uh is an analytic function of b and of the temperature so given the temperature uh and given d we can construct a matrix b and once we construct the matrix b the probability distribution is gaussian as represented here is a gaussian centered around rc with an extension which is given by this b matrix which is analytic in the energy now we can use these to understand the meaning at least of this rc indeed if i now i consider as observable of r the position r of a nuclei if i do an average of a nuclear position with a gaussian i get nothing but lc so for better reason i call r the centroid because this represents the average position of the of a nuclei average over the the far right temperature distribution which contains both temporal temperature and quantum situation so this is the so rc as a meaning which is very quantum centroid which is something that for example we measure with diffraction if we do a diffraction experiment what we measure in diffraction experiment is nothing but these these uh centroid now uh let's look so so we found what is the meaning of rc now let's see if d can be associated to the phonon that gets soft as a phase transition now we have a condition is right to do the shaman minimization the shaman minimization is meaningful just if this distribution probability distribution is bounded so this means that can be integrated and is bounded only if the matrix b has eigenvalues which are all positive and one can show that if given the analytic form if the eigenvalues of b are positive also the one of d must be positive so this means that the condition that the probability is bounded implies that all eigenvalues of b are positive and also all eigenvalues of b are positive if all the eigenvalues of b are positive this means that cannot go to zero and this means that these are not the physical phonons because we know that we saw experimentally that at the phase transition the eigenvalues of the real phonon became uh zero they soften because they go to zero so this means that d so rc has a meaning which has the position that we measure with diffraction but d doesn't have any meaning now uh so how we go beyond that so to do that we have to uh to consider the uh land-down theory of phase transition so in this case uh we given a phase transition or for example a ferroelectric or a tendency way phase transition we need to define an order parameter of a phase transition and the order parameter is what for example we measure by diffraction so I told you that if you consider a high temperature phase of a chart this way or a ferroelectric phase what we measure by diffraction we measure an isometric phase by diffraction we measure much more than the centroid of the of the atoms and so the centroid are the order parameter that define in which uh the phase in which we found our self so which define the phase transition and so uh so in this case we should uh using the land-down theory of a transition they find a free energy which is a function of the order parameters that fix the phase transition so in particular are uh in the in the case of uh our ferroelectric or chartness to a phase transition are the centroid position and how we can do this is sufficient to define the free energy from our shaft free energy minimizing the free energy with respect of uh the mannotta with respect for a given rc so we do if you want a constrained minimization in which for each rc position we minimize with respect to the d variable so in this case we have a free energy landscape which depends on the average position of the nuclei and at that point we can define for example the condition occurring uh at a second order phase transition so a stable phase what it is a stable phase is a phase where the first derivative of free energy with respect to centroid is equal to zero and the second derivative of a free energy with respect to centroid is positive definitely and so uh this second derivative in principle depends on temperature or on pressure for example if you play with uh with an unbiased pressure and what we have is that the phase transition is the what we have is that the eigenvalues of this free energy curvature acquire is the moment in which acquire a negative eigenvalues so the eigenvalues go exactly to zero at the transition temperature and below the transition temperature became negative and so what we have to do is to compute this second order derivative of free energy with respect to to centroid now can we do uh with respect to the analytically in the context of a second thermonics so this has been done in this publication raffaello pianko the major actor of this of this paper and raffaello what he did is consider indeed the uh second order the second order uh the curvature of the free energy with respect to the centroid and uh the nice thing is that uh we have an exact result uh considering the approximated shaft free energy so uh the shaft uh free energy has an exact expression for the for its curvature and this curvature is equal to the d which is our auxiliary matrix which is always uh positive dignity but in addition to that we have another terms and these terms has a mathematically this form so we have a rank free tensor so uh this means a tensor which depend on free uh atomic indices here is a rank for tensor here another rank for tensor and this is another rank for tensor and then we have to do a scalar product contracting two indices of the tensor here two indices of the tensor so at the end we remain with an object of containing two uh two indices which looks like a a a a dynamical matrix if you want now what are these these uh these tensor these are these special tensor that can be shown can be it's nothing but the average can be written also as the second derivative of the uh bonopenheimer potential in the surface average over our gaussian uh distribution and this rank free tensor this rank for tensor are analogous of this but taking free derivative of all position in average it over the gaussian uh shot the average distribution and these are the same things so this is written in a matrix vector form now let's express it in terms of uh uh green function uh what we have is that the curvature looks like the auxiliary field uh dynamical matrix plus a self-energy where the line here corresponds to the phonon green function with a auxiliary Hamiltonian and the vertex we have a free phonon vertex a full phonon vertex which looks like the average of these uh for the potential this for for the potential here we have in summation so there is like a chain of diagrams in this matter the good news is that we can evaluate this object numerical for realistic system so let's look at the to the application and so now i will before to show you how i can we compute both the the free energy and this object i show you an application and then i go back to the methodology and so for that i consider the two uh trans density ways so we did many applications but i consider i focus here on yobu the selenite which is the system i already illustrated to you and an analog system where in place of selenium i have sulphur atom and what we have is that uh for t below if if a Chinese wave exists for for in the low temperature phase we have a low symmetry phase we have a secondary phase transition at t Chinese t wave and we have a Chinese with melting in the sense that we have high symmetry phase for the larger landings and now in this part the question i want to answer is the following uh usually charge density waves are considered as originated from an electronic electronic release of freedom so from electrons and so the question is that is this melting so is this uh temperature with temperature as an electronic origin so originated from thermal excitation of electron or this is ruled by quantum and thermal an harmonic fluctuation of the nuclei so the theory i developed so far is to consider the second effect but after if you look to textbook the scripture of Chinese t wave often the the melting is attributed to a mechanism where the an harmonic fluctuation of nuclei are are neglected and the effect of melting is attributed to the electronic release of freedom so the things i want to show you is i want to compare these two pictures for these two cases okay so let's start so i started with not uh selenium but with sulfur the obviously uh sulfur too now what is the experimental situation the experimental situation is that if we replace selenium with sulfur we observe no charge density wave for the bark but the very final still show a very strong temperature dependence so in this case i would say that people argue that is an incipient charge density waves so we are very closer to a charge density wave transition but the system doesn't make it so it's not able to do a charge density wave whereas in the monolayer there are conflicting experiments because in some experiments on graphene in the monolayer a charge density wave ordering has been observed but not for example from gold so let's consider this case so again the phase is identical to the one of of uh sulfur that i already show to you uh so here i show you how atoms are arranged so uh we have again the transition metal atom which is in the middle and we have uh free triangle on top which constitute the calcule dynamic atom on top and the bottom this is a in absence of of a charge density wave and then we have a different arrangement the most energetic favourable one is between each phase what the arrangement is described here okay so uh let's look at the experiment into the phonon so the experiment again i would say in the lattice scattering experiment and measure at two uh temperature at low time at high temperature is the uh right the points here here and here and uh if we cool down the system at two kelvin you see that there is a soft a very strong softening of this phonon branch so the phonon frequency here is reduced by a factor of three or even more so a factor of five so there is a very strong temperature dependence even lower in just between 300 and zero kelvin but you see that the phonon remain at the finite frequency so the system does not give rise to a charge density wave to a charge density we need to touch the zero energy axis now if we do the harmonic phonon these phonons be computed with quantum espresso is the standard harmonic theory quantum espresso and uh you can see we obtain the uh these uh dotted lines and so we obtained that phonon indeed here below are not negative but imaginary so we obtained that phonon the system is not stable harmonic system is not stable so uh the system lacks of stability and uh an harmonic calculation would predict a low a distorted uh for the lowest energy phase is a phase uh with a reconstruction actually in this case a two by two charge density wave reconstruction now let's look at the effect of electron temperature of the temperature so here again if you look at the phonon at the this i think at the end point actually not the end point closing this point if i take this moment at 0.75 gamma m and these are the experimental phonon frequencies square i do the square instead of the phonon frequency in such a way that the experimental phonon are well described by linear behavior and you see that it is a very strong temperature dependence but as you can see i don't touch i don't arrive to zero so therefore no remind finite even at zero okay now i can do the same things uh plane computing the harmonic phonon as a function of electronic temperature using quantum espresso as you can see i have a linear uh behavior to characterize the again manager square the only things that that in this case i get something qualitatively similar to what observe in experiments but you look there are two uh problems the first one is that in this case i touch the zero axis as i already show you the plane of transparency at zero temperature harmonic phonon are imaginable so uh the system is no more stable and the other uh things is that here i have a scale of 1000 of uh kelvin and here i have scale of hundreds of kelvin so we miss both the stability at zero temperature and also we miss the behavior so that we need to increase the temperature by one one order of money to the compare to the to the experimental one now if you do the same things but with the anonymity so this is the result of the shock calculation and the again the dotted blue and the dotted red are the experiments and the the black harmonic phonon and the blue are the shot theory uh at uh uh zero temperature three are the kelvin you see that we indeed the phonon are stable and we describe such stronger uh phonon temperature now i think i'm a little bit longer so uh probably the best is to uh no shall i cut it down or i don't hear you you have you have 10 more minutes or so okay okay okay uh so uh in the case of monolayer as i told you in uh let's go to the monolayer in the monolayer uh on gold uh on graphene here is the stm image that identified a three by three reconstruction uh at low temperature so uh there is a channel st wave in this case and whereas on gold uh there is not uh this is on gold this is not observed such a reconstruction so on gold the monolayer observe a channel st wave and we do uh the shock calculation on uh on the monolayer at the experimental lattice parameter we see that before we get soft but you see that here is a function of temperature but and this is the harmonic one but in the shock phonon we never touch the the zero energy axis so it can be stable but if we uh expand the lattice by five percent the phonon gets soft but this means that the we are very close we are going closer in the monolayer to the phase transition and just plainly to be there with the parameter like the lattice parameter also the doping we can make the phonon unstable and that is compatible with what was observed now let's look to the other case as a bulk monolayer in selenite so again i don't repeat that this was the description of the statement so before so we can look at these i want to use this data now let's take this data and and consider the frequency consider the frequency square in such a way to have a linear slope which is here so these are the experimental data it's the same of these but here i present frequency versus temperature here is is frequency square versus temperature in this case i get a line with intersect zero energy and 33 Kelvin and the first things i just consider the harmonic phonon increase the temperature and observe something similar to what observe the other compound so observe indeed the stabilization by internal temperature but the i need to go to a much higher one other of mine to higher actually more than one other of mine to energy scale to to reproduce the experimental data so again in this case the internal temperature has no role in the phonon hardening whereas if we do the shock calculation which are the blue line and purple line different color line we again observe a behavior close to experiment and these are the result due to ionic situation compared to experiment this is just a zoom in this area as you can see that these are the experiment this is the theory for the bulk we measure 60 Kelvin which is close to 43 looking especially to a large energy scale and for the monolidia we observe a temperature which is similar so in this case we have a weak dependence on dimensionality and we produce the experiment now the last part is how i did this calculation now this calculation has been done with a stochastic approach the stochastic approach is describing all papers and recently we just published to publish the code the code this is the website of the code and this is the paper associated to the publish code where we revise all of our latest implementation also latest algorithmic and and the code is interfaced with quantum espresso can use some user quantum espresso as a force and total energy engine but we can also use other principle code like a vaspe or any other code even if the main most of our application done using quantum espresso i think the time is is short so i don't have time to enter in the in the discussion of the details and let me just give you the feature of the code so yeah the computational cost to do a shot calculation comparable or smaller than uh uh company nanotype and decalculation so we need uh basically for each some decalculation in supercell and uh we need a few thousands of configuration to converge the result we include the thermal and quantum filtration is a variational approach so we can deal with larger than monicity we've uh have access to the stress tensor so we can do full self minimization of the crystal uh we have a direct access of the free energy without need of thermodynamics indication scheme that need to be used for example in uh molecular dynamic type of approach to extract the energy and this is great because we can compute the temperature of first order phase transition we have great access of second order phase transition thanks to the free energy curve that we just described and the approach is based basically on a Monte Carlo type of stochastic approach of minimization of the shot uh finally i want just to point out that what i presented to you is to do thermodynamics not to deal with so to deal with transition temperature reference transition but we are interested also to describe uh spectroscopic properties so in this case we have to extend the basic consider harmonic approximation to time domain like we need to extend the density functional uh that is able to describe current state property to time dependent density function we want to describe uh the excitation properties and so we did this actually uh Lorenzo Monacelli did this in this very recent paper uh so we extend the to time uh domain and this is able with this we are able to compute a petroscopic properties like Raman infrared with the inclusion of an harmonic equation again i don't have time to enter in that and i want to stress that is not with patina and Monte Carlo it's not possible it's possible to compute the harmonic properties but not time dependent properties okay with that i put my conclusion so i show you that the uh centroid so the other atomic position is a good order parameter to study second order displacitive uh phase transition like a charlesley wave or a ferroretti transition but the curvature of free energy change sign at the phase transition we obtain an exact analytic formula for the free energy curvature that we can evaluate numerically in both neobium diselenite and neobium disulfide which are density instability is related to state which are close to the thermal energy and better we see because by eating with electronic temperature premium temperature eating at thousand of Kelvin which is this energy scale we remove the charlesley wave but the main thing of serving experiment is not related to the electronic excitation but is related to the ionic and harmonic quantum and time of fluctuation and which are described by the psychosynth and harmonic approach that i described to you and in the case of the monolayer we have that the environment plays a crucial role in the changing the lattice parameter and in doping we can change the transition temperature of the system okay sorry to be too long try to be deductical okay so i'm open for questions okay professor francesca maury thank you very much for the excellent illustrative uh lecture and now the uh the question is presentation is open to questions so do we have a do we have some questions so now i'm trying to unmute celebrate the eyes this is stefano please stefano celebrate the eyes thank you very much francesco for this spectacular talk and impressive results much appreciated thank you for being with us just a very general question i appreciate your remarks about the different scope of of the self-consistent harmonic approximation and path integral molecular dynamics or montecarlo whatever it may be concerning dynamical properties and spectroscopy so this is uh received how about uh equilibrium properties how would the limitations of the two approaches compare both conceptually and uh in practice what path integral or md yeah path integral and self-consistent harmonic concerning uh equilibrium properties uh so uh now the path integral montecarlo is an exact approach if you want to like md but this advantage of path integral montecarlo is that uh we have a the complexity of a calculation in crisis lower in the temperature so if we go uh to t qual zero the computational effort in crisis decreasing the temperature so a very low temperature became extremely expensive and it's much more expensive than uh classical molecular dynamics so uh so the montecarlo uh so i think that is something that we are you know first second limitation of montecarlo is that we don't have access of both md and montecarlo we don't have access to the free energy directly but we need the the thermodynamic uh uh integration so the the the computational cost which already a larger than uh of to say the tower meter is comparable to md but montecarlo patina montecarlo is a we need to add many replicas of the calculation which could be one hundred one thousand depending on the temperature so it could be one hundred one thousand times more expensive and uh molecular uh dynamics in addition we can recognize md and montecarlo need the thermodynamic integration so let's make uh our approach less expensive for the computational point of view from a point of view of accuracy obviously md and patina are exact so is a is a so we need to do a comparison between uh the two is something that we are doing right now so we are so i don't have a direct comparison to present to you right now obviously uh our goal is to compare so in our calculation we can put h bar divide the best root of m going to zero and obtain a big uh md result and so we can compare uh with md which is for which we can have a much more precise number to compare with and something that we are doing right now so i don't have a direct comparison uh on that uh instead one things that cannot be done at all can be done in monocular dynamics is to do time uh correlation monocular dynamic allows to do uh heteroscopic calculation and so again we are right now we are benchmarking the approximation result of approximation against molecular dynamics putting h bar to zero in this case we should get md is the uh if we want the exact benchmark to reclude user with advantage of it with uh we can also consider h bar from zero in the case of quantum uh system is not possible uh or either it's possible but it's extremely complicated for a spectrum it contains more than one peak to obtain something with but in the ground so i don't know if i gave you an answer there are cases where uh so one strong limitation of the chart for example if with rotation it is your freedom for example if you consider the phases of hydrogen and the pressure there are uh phases uh high temperature where the hydrogen dimers are free rotator or quasi free rotators a rotation is not well described by a cartesian a Gaussian cartesian coordinates and so this case is the case where the shot fails and but we are working on data so we are uh these are students following the school with uh these phases actually is to use a curvy linear coordinates to include the rotation of quasi free rotation this uh uh just uh i'm i'm a bit surprised because i think it's not just a matter of uh curvy linear versus uh cartesian coordinates it's the fact that uh rotational motion is uh is diffusive and you cannot do any diffusive thing in uh but if it's very diffusive so depends if it's a melting so totally totally a total free rotation will not be able but the total free the zero point motion of a total rotation goes to zero so if one of the contributions to the free energy is very small obviously it's not so interesting in this case so the most interesting case is if the case where we are quasi free rotation so or some degrees of freedom are are melted and some not the melted one will contribute with a very small contribution to the free energy of zero point motion and so we but if instead if you use the shot better we became harder for the frequency so we do a bigger mistake so this is a case for example where the shot right now doesn't work for molecular crystal where uh rotational motion are present because the shot uh make it finite uh this mode at finite frequency so they give non-negligible contribution to the free energy and so the free rotational phase the free rotation phase turn out to have a very high free energy compared to the exact one thank you very much grazie grazie francesco okay thank you very much for the extensive answer until we have a new question so nougra you can ask the question yeah thank you professor memory for the great presentation and sorry i might have a very basic question you mentioned somewhere about the isotope effect on the phononic properties and i'm just curious if the isotopes will also change or modify the electronic properties for example suppose we have graphene consisting of all carbon 12 and another one consisting of all carbon isotope with mass number of 13 will there be a difference in its electronic structure yeah so the the method that is done to treat the ionic degrees of freedom so and and there are many manifestations of this for example if you consider for example if you consider eyes the transition so there is a transition between phase so probably you saw this because it was described by roberto car last week there is a very the structure of phases i-7 and i-8 where the molecule has an hydrogen bond and i-10 where the hydrogen sits in the middle so this phase transition between the transition then we have a well defined molecule and i-10 where the hydrogen sits in the middle of there is a shift changing the hydrogen with deuterium of of ten giga pascal so this is a shift of ten giga pascal between one of the other and this effect is an isotope effect due to the zero point motion of the nuclei indirectly these are effect on the electronic structure one example one crucial example for example is h3s which is 200 Kelvin superconductor also in this case we have a shift in this case of about if i remember well 40 or 40 giga pascal about two phases and so the tc is high just in the cubic phase but in the lower symmetry phase or visual phase tc the transition temperature is lower by a lot and changing my isotope you favor the original phase so there is an impact a very strong impact on the electronic properties through the structure and the structure is impacted through the zero point motion and this one this kind of impact is described that we have the paper and nature paper similar things on lanthanum h10 lanthanum h10 is a is a superconductor superconducting at minus 20 Celsius so very close to one temperature also visa there is a stabilization due to zero point motion of 100 giga pascal in this case so it makes stable the high symmetry phase which is superconducting 100 giga pascal smaller than in the case where the zero point situation is present if you do with deuterium this stabilization is much reduced I don't remember exact number right now but also that can be published in another paper and we would dismay visa this approach to computer okay thank you clear okay thank you very much and now let me just convey that a lot of people are thanking you for a very nice talk on the zoom chat and then we also have a question on the zoom chat and let me read it and it says if negative frequency at zero k is shown then does it show instability of structure or it is just due to parameter like temperature or electron interaction no way so there are two kind of it may I show you two kind of stability let's see let's see here for example or here so this is the neobium discelling on two so these are harmonic calculation obtained in quantum espresso if you do the calculation neobium discelling in two in this school on this structure with quantum espresso if you do at zero temperature which has been reproduced by makes it makes an impact on type of smearing cold smearing you obtain this black line this means that the potential is a negative curvature and so if you displace the atom from equilibrium you relax you put the espresso you find a distorted structure so this instability corresponds to instability of the energy profile whereas if we do here our shock calculation in this case the instability is not with respect to the ionic position but to the average ionic position and the instability is not of the energy but of the free energy so it's the free energy with respect to the centroid so to the average position and also in this case if we do if you do a shock calculation relaxing all the parameters and displacing the atom from equilibrium the shock order the shy interface which is publicly available will give you a relaxation of the average position with a supercell reconstruction so in both cases the presence of imaginary frequency corresponds to instability but this is the instability of the average position of the free energy with respect to the average position whereas here is the instability of the energy with respect to the not the average but the exact position so there are two instabilities quantum espresso can do the minimization on this and our shock order can do the minimization on this using quantum espresso as a force and total energy engine so there are different examples in our distribution but interface also with a quantum espresso code i don't know if this clarified okay thank you very much uh so the the person who asked the question is uh said thank you thank you sir and now we have a question on uh slack and let me read it it goes since self-cosystem harmonic approximation is a stochastic method how to generate a new state ionic positions what is the interior to accept the new state yeah so if you want uh i have uh four slides on that then depends on much time and we have yeah please please go ahead okay so as i was the part of the talk that i i i skipped i skipped because of time so so uh so this is described what uh you find in this paper and in this because you can download so what we need is that basically this is our uh free energy that we want to minimize and the parameters are the parameters that enter in in the triad amiltonian so are the centroid position and and the uh curvature of the triad amiltonia not to be confused with the curvature of the free energy so basically uh eventually we are so here for simplicity i don't describe the case where we also minimize the cell but in principle we can not in principle in practice we minimize also maybe we can sell this case beside these two parameters we also have the cell parameters so now so this means that uh in minimization we do we should minimize with respect to this set of parameters and minimization what it is is nothing but the trajectory in the parameter space so we start with a starting guess and initial which is called zero friend then we have a trajectory like we go from zero to one which is the configuration to one to two etc till we arrive to the converge values of rc and d like minimize the energy function so the strategy is to use a minimization based on the gradient in particular we have a country gradient minimization and to add the country gradient minimization we need to compute the gradient of our free energy with respect to these parameters so so how we do it and we do uh thanks to the fact that you have an analytical expression for the gradient so let's consider for example the gradient with respect to the centroid the gradient with respect to the centroid is equal to what will be equal nothing but an integral over the probability distribution which we want to make as I described you before so is this is the diagonal part of the density matrix which is a gaussian centered around rc with a curvature which is an analytic function of d and p so for even each d that we have we have this with gaussian distribution and then the forces are nothing but so the this gradient is the average of the difference between the dft advantage of forces so the one provided at quantum espresso minus the forces given by the trial harmonic Hamiltonian which again that depend on on on d and on rc so this is a analytic function of d and rc and this instead we use quantum espresso so we need to evaluate these in similar manner the gradient with respect to the the matrix is again the integral over the this gaussian probability distribution of an object which looks like something that's similar to what we find in the theory so we have again the difference between the the ab initio forces and the harmonic trial forces and here we have a difference between the position and the centroid position and then here we have the matrix which is a matrix which again is an analytic function of the at okay so we need to evaluate this integral if you have this integral we have the our gradient so how we generate this gradient you see that all our integral has this form so is an observable of the before of the ab initio forces where the observable is analytic a part of the ab initio forces times a gaussian probability distribution that depend on rc and d and the initial step will depend on rc and d of the of the first guess of rc and d so how we do so we start from a given rc and d which is the zero starting guess given the fact that we have gaussian we can generate an nc ionic configuration according to the probability distribution of the initial distribution so this is gaussian which can be analytical this generation so we generate with a random number generator this we display the datum configuration according to this distribution we do this for a supercell so on this of the calculation like in md and then we substitute this integral with a stochastic evaluation we just do an average of our observable we depend on the on the displacement position our i generated with that the datum variable configuration so with that we have the gradient and we can compute in the stochastic way the gradient now that's good the only thing is that once we compute the gradient we can compute we can move the configuration along this gradient and to the first configuration but now if you do the first configuration we have to generate anything from scratch and we lose all our expensive gift calculation if we lose our expensive gift calculation we have to do all the calculation from scratch but it's bad because eventually this configuration is not very different from the initial one so we want to recycle somehow the fact that here we did the initial calculation how we do this we do with the regular thing so we still use the n ionic configuration generated with the zero the first strayal configuration and in this case at this point we can evaluate the gradient just if i divide the if i do the same summation but i divide by the probability to generate the configuration in the position our i at the time zero and multiply by the probability to get the time i this is a Gaussian this is a Gaussian i have an analytical expression so this ratio is analytical and so we think i can continue obviously the more i continue more so in the beginning this ratio is equal to one if i continue more and more this ratio will be always more away from one and the noise or stochastic noise will increase at a given point this is just noise so i will have to to to re-concute everything so this is the algorithm so basically i generated our i according to the initial probability Gaussian distribution this is the only cpu intensive part so i use their initial engine to compute the forces on this displaced ion configuration i use important sampling and i wait to compute the gradient of the free energy with respect to both our c and d i use the gradient to update to obtain the new values of our c and d and then i see if the the new configuration is too far or or not from the initial one so if this ratio is close to one it means that it's not too far so i can just recycle and so i do some loop here without calling anymore quantum espresso and this costs nothing instead if this goes uh is too far from one then i accept that the new stuffing configuration equal the one at the minimization i go here and so this is the minimization this is an example for for a crystal so this for example is this ratio i start from one this is the free energy so i minimize the free energy here at this point i go away from too much from one so so i re-compute so i call again quantum espresso i go here etc etc and during the minimization i start with a black line which are the dotted line then here in the middle lies the red and then at the end there is the curvature so this is the way and to compute the curve i should admit also to compute these expectation values the third derivatives now quantum espresso unfortunately is very expensive in quantum espresso to compute the third order derivative we can use it one plus one but that is a very expensive part of the code and there is no something able to compute this for all the derivatives so we cannot do this average in this way but what we can do we do the average of data which contains three derivatives but the average is multiplied by what by a Gaussian so we can just integrate by part so in integration if we do the integration by part i can remove the derivative from here and put on the probability distribution of the Gaussian and if i do twice i can remove two derivatives here and these are the one that are the what's result in the derivation of the probability Gaussian probability distribution so i get the b matrix which is what's entering the exponential and so removing two derivative by i have here the the Gaussian and i remain with the first derivative which is what's providing the quantum espresso and the same she can apply for the third derivative in this case i integrate three times by part and i get an expression to these which has the same form of previous one so this form of an observable that depend on the only position temperature rc nd and the ab initio forces and again i use the same trick also before so this is how the stochastic approach is applied and all these is publicly available and can get there on the side okay thank you thank you very much and maybe now the the quick answer to the last question which is from the slack so the question is the following how to calculate the charge density periodicity from the phonon spectra at temperature which is lower than the critical temperature they so so what we do in the show we do as i told you we do supercell calculation so for example for this the newbie on this night we have a six uh by in the i think a six by six by one supercell and so that so in principle the calculation is exact only in these uh in the for the momentum the quasi momentum commensurate to a six by six supercell and so uh if we get the ones of these uh momentum that goes imaginary that is the charge density wave momentum then what we do like in quantum espresso we can do for interpolation and getting something close so if it is the grid is dancing up we can see also the charge density wave at the moment which are not exactly at commensurate q-point indeed here the full lines obtained by Fourier interpolation so and in this case we can identify the point so for example this momentum is the momentum of the charge density wave which is not is not necessarily commensurate i have to say that to do that anyway the interpolation should be done with a pretty large supercell so in order to leave a major limitation of of the shot it's the part too that we need to consider very large supercell if you want to describe the exact momentum and if you want to converge it to the system size to the system size but this is a common problem also of molecular dynamics and patina montecarlo in molecular dynamics and patina montecarlo unfortunately also we need to use supercell and we have an exact answer just with a momentum commensurate with the refolding of a supercell so that is a common problem we can interpolate the Fourier interpolation of the not the final result about the intermediate result is something we are working right now so i think i think what this is still so if you do harmonic calculation the interpolation works much better so this is harmonic calculation in this case we can do all the calculation with unit cell and with a large enough grid of full momentum point we can we can basically arrive to a thermodynamic limit in patina integral molecular dynamics and shot we still need the 12 supercell so we have less power for the moment okay thank you very much so i think that that now is really the time to close this session and so if there will be some more questions so you can post them on post them on the slack and also on the youtube channel i think it will be still open for some time so at this point i would like to thank again to professor maury so thank you very much okay thank you for your attention for the invitation thank you thank you very much so what i should go the nice things is that this is not more an answer but is an exact result an exact dice on equation if you want and we can see for example i'm distrusting in the in the in the perturbative limit because it is interesting if we do i don't know if some in the in the course did the perturbative theory of phonons in the perturbative limit so if we can consider the the Taylor expansion in terms of the of the third leader we give free foreign vertex for foreign vertex the lowest uh uh green uh the lowest order the greater the function is equal to the harmonic green function plus the tappel diagram the loop diagram the bubble diagram here i have to stress that this is the third derivative of the potential at the minimum whereas visa is the third derivative of the potential average over the gaussian the gaussian uh situation so this vertex is different from this vertex obviously in the limit of of the gaussian width going to zero the two coincide same things for these vertex and if one of the interesting things is that for example the green function with the auxiliary field in the productive limit give the tappel envelope whereas visa give the bubble so you see that the the what's it's found in perturbative theory uh enters in different part of its shop the nice thing is like that visa can be uh is a can be made rigorous in the has been made rigorous in visa in this publication and in this publication basically we have a a way to compute the autocorrelation function or any observable dependent position and now if the observable is the polarization we get the infrared spectra if the observable is the polarizability we get the raman spectra and but we can for example if the the observable is the thermal current we get the thermal conductivity using the coco formula so it's a nice way because it's a way to compute all time correlation function of any observer which depend on positions okay thank you for the very detailed answer if i may i would like to raise a question maybe conceptually if i write what i catched so when you propose to to introduce a free energy and you manipulate it with the unharmonic extension indeed we work with the system at a finite temperature we immediately induce let's say in a instantaneous normal modes negative values but the question is how to find the transition in the present of finite temperature and not associated due to the dissipation through the temperature and if we take a look at the liquid and start to cool whether you could expect using your approach to find something kind of glass transition because you free your approach provide to avoid the dissipation through the instantaneous normal modes because it's hidden through the apply of free energy i hope my answer is the point is better so for the moment all the system i've described to you are a system where we can define centroids and the system make fluctuation around these average position a liquid is a case where this entire does not exist so that's not present so it's a little bit so it could be that if you want it's true that in a liquid we also have a so when we have a diffusive a diffusive behavior we cannot describe it as a as a gaussian situation around equilibrium position obviously in a liquid we can think about the average position our slow degrees of freedom that diffuse and a shock can be used to compute the choppy part and the quantum visceral point motion of the fast digisofenol but one has to find a rigorous way to divide up the slow and fast and it's something we didn't treat so far so i would say that shaft remonde as we develop is not meant to consider a order to disorder but just order to order phase transition okay but it's interesting to to explore this yeah but what you said that it maybe we should start it from the glass but go into the liquid and then central is lost to the shape and again or we could indicate some transition you can see a mechanical melting so for example imagine that i take a i overheat a crystal so this means that usually the melting is the first order phase transition so i can remain if i heat up slowly so imagine take ice and if i take ice and i a full piece of ice with no nucleation site i stop above zero celsius actually the ice remains stable locally and so we will not melt till i write to four celsius five celsius at a given point the system even if there are no seeds for the melt can have a mechanical breakdown and mechanical breakdown is when the is no more mechanical stable so so in the sense that ice is a local minima till i give a temperature above that temperature is not even more than local minima shark can compute the temperature at which i lose the local minima and so in which the system became mechanically unstable and and but also we have examples in the in the in the paper that i show to you indeed in this case not to a melt but it is it preserves a crystal to crystal phase transition in which if you do a shock calculation we have an hysteresis and the two uh and two limit of hysteresis temperature so uh are the the the hysteresis cycle is limited by the two points where the crystal became mechanically unstable uh then uh yeah so the uh the melting uh is the first order melting uh i don't know so to what we liquid the first order to iniquity that i don't have any idea if we can if it's the best method to deal with it or what we learn from from so we didn't start so far okay thank you very much okay and now it seems i at least i cannot see any further questions on slack and it also seems there is no more questions on the zoom okay so thank you much probably by now you are very exhausted okay it's okay okay thank you again for the for the interest okay bye everybody yeah bye bye