 So I guess you can hear me and you can see the slides. Okay. Yes, so I'm John Kiriaco and I, today I want to tell you about the work I did, some of the work I did during my PhD at Columbia. And particularly this work focus on the voltage, the driving of the matter into the transition calcium root innate and how this can be explained by hitting effects and affect the effects. So just to give a bit of an overview and introduction to the topic. Now matter into the transitions have been very studied in recent years. And in equilibrium they can be driven by many tuning parameters like for example, temperature, strain, doping, and many more. Like you can see just an example in this space diagram here, which is for calcium root innate doping strontium. And depending on temperature or doping you can have many different phases in this material. This is just in equilibrium. But then recently there have been many experiments performed out of equilibrium. So in which the certain material is, the transition in certain material is driven through electric currents or electric fields. And you can see this in the phase diagram always for calcium root innate material in which you have a tuning parameters, the temperature and the applied current. And you can drive many non-equilibrium phases through the application of this current. And now these experiments have been focused mostly on inducing the matter into the transition in multi-insulators. And there was one example, which was famous some years ago, which is the experiments performed on a vanadium oxide VO2 in which initially the transition was thought to be due to the driven transition through the electric field. Initially it was thought to be due to purely electronic mechanics. So there was something at the level of the electronic structure of the material which was changed by the electric field. But then it turned out to be just heating of the system caused by a dual heating of a dual electric field. And you can see just from this table how many materials have been studied, how many materials this made the insertion transition driven by fields as been studied. And what is peculiar about this material about calcium root innate is that the threshold field, a breakdown field required to switch from the insulating phase to the metallic phase is very small. It's like three or four orders of money are smaller than what happens than the field required in the other material. So during my PhD I decided to kind of focus and try to understand what is going on in this material. Now understanding how this non-equilibrium phase transition work is also a challenge not only from the experimental point of view but also not only from the point of view of modeling what was exactly happening in the material but also on a more theoretical standpoint of view. So like the first issue of course is to understand whether the mechanism that is driving the transition when you apply the electric field to your material is just electronic in nature. So you are messing around with the structure of the electronic model or just heating up the system. So like your sort of pseudo equilibrium transition which you just raised the temperature of the system. And then more on the theoretical side there is the question of how we studied this electric field driven state and whether we can use some sort of pseudo free energy functional to study the non-equilibrium transition similar to what we do in equilibrium with the land-off free energy to obtain the various phases of the system. Now in this work this work will be divided mainly in two parts. So first I will be talking about some possible microscopic theories of this transition from each other to metal driven by an applied voltage. And then I will focus more specifically on the case of a calcium rate and how this can be explained by some Peltier and thermoelectric effects. So more on this theoretical models there is the question like how does the current or field applied to the system actually drive the transition. So the usual models we consider are systems with a certain bandwidth W, a gap delta. So there is an incident phase and a metallic phase and the incident phase has certain gap delta. And usually there is also like some interaction strength because these are correlated systems. So there is a competition between on-site interaction between electrons and then hopping so kinetic term. And then there is temperature T and all these parameters that they mean the equilibrium phase diagram. And the systems usually have this incident phase with a conduction balance band and they can be studied by different techniques. So one way to study the systems is to model them using the Hubbard model and then apply for example an equilibrium dynamical field theory. So this more nomadical approach. Then there are more analytical approaches in which these systems are modeled with the samtoy model. So for example, systems with the antiferromagnetic or charge density insulating phases. And then you insert also in this system some relaxation processing and then apply in field and you try to get some analytical results from these models. And of course as I was mentioning there is this theoretical issue of this non-equilibrium to the free energy which is not clear how to define. Like it's not clear exactly how to define this to the free energy and what are its properties while it's of course clear for the equilibrium of free energy. So now for what concerns some possible driving mechanism of these transitions. One for the simplest one is just studying the formation of the energy bands due to the presence of the electric field. So there is these electric fields and then the energy structure of the system is changed. And this can lead to just like a closing of the gap in the solution phase. So the system becomes metallic because you're no longer have a gap in the electronic structure. Then you have the mechanism of Landau-Siener tunneling which is just a tunneling of electrons from conduction band to valence band. So basically you're exciting some electrons and now are free to move and free to conduct. So like you have, it's possible like you through the electric field you induce a current and then this leads to this stabilization in the welding phase. And then there are heating mechanism which you can only heat the electrons of the system or you can also heat the electrons and the lattice structure. So you're just heating the entire system. And these two mechanisms can be different. So there is like some subtletying when they are important. And for example, if the order parameter associated to this transition is pure electronics is just related to the nature of the electronic phase, then it doesn't matter whether we are heating up only the electrons or both the electrons and the lattice. But for example, if there is some couple of order parameters of the transition, so like if both the lattice and the electrons of your system change the transition then it's possible like only heating up the electrons is not enough to the transition. So you also need to consider heating up the lattice. So there is this slight difference that can be important in some cases. So just to give an idea, when we talk about the deformation of the band structure, like from a very intuitive point of view, we can have a simple picture in which, so if you consider your system, your lattice made up of different unit cells with a unit cell length of A, then electrons on the membrane unit cells, membrane sites have an energy difference approximately proportional to A and then times the electric field E and then the electric charge. So when this energy difference is proportional to the gap with the incident phase delta, then you're basically closing the gap with the incident phase like you make the conduction balance band touch. So you're essentially transforming your insulator into a metal. But this effect usually requires very large fields like the order of 10 to the 7, 10 to the 8 volts per centimeters. So it's not really the mechanism which is going on in many of the materials or in calcium because in these materials the required fields, the threshold fields are much smaller. Then there is the mechanism of Landauziner tunneling in which as we're saying, there is the possibility of this tunneling in between conduction and balance bands, you're just exciting electrons across the gap of the incident phase. And in correlated materials, usually when the gap depends self-consistently on how many particles you have in conduction band and balance band. So if you are exciting electrons to balance band then basically you can reduce this insulating gap so you can break down the insulator. Now this in the Landauziner tunneling, the probability, so there is a rate which is proportional to the electric field E, then this rate is multiplied by the tunneling probability which of course is suppressed exponentially by how large the gap is and in particular it's even by this formula here. So you have an exponential which is exponential of minus the gap squared over the electric field. So this is the quantity that tells you how important this mechanism is. And usually the threshold field required to have this Landauziner breakdown of the insulator is 10 to the 5, 10 to the 6 volts per centimeter. So it's better than this mechanism here, the study of the deformation of bands. But it's still a very high field. But also it tells you that insulators with small gaps are easier to switch. Now I want to show you some more detailed models of like for example, correlation systems in which there is a breakdown of the insulating phase caused by electric field. And this is a model from a work by Mats Amarici Capone Fabrizio. I was done here in Trieste some years ago. And basically what they do is they take two bands of the model. So they have the usual hubbub model for electrons and then they consider electrons living on two bands so that they can have a coexistence of insulating and metallic phase depending on the interaction strength of the model. So like for a certain random interaction strength there is a coexistence of insulator and metal for large interactions. There is an insulating behavior and for small interactions there is a metallic phase. And what they find is that when you start only from the metallic phase, then when you start from the insulating phase then the insulator can collapse thanks to Landau's inner tunneling. So they find that this mechanism actually can collapse the insulator. But then they also find that in the coexistence region you actually need smaller fields because when you have this coexistence, so these two phases coexisting, they both elect some free energy. So you can draw the free energy depending on the order parameter. And what you need is for your electric field to just modify the energy of the two phases enough also that like the metallic phase becomes favored compared to the insulating phase. So just a sort of a free energy and immunization balance. And they find that like in this case you don't need the large fields required to have Landau's inner breakdown but you can have a transition for all smaller fields. Then there is another model in this like more recent paper by Matthias and collaborators in which they again study hardball model using dynamical field theory and they have an insulating phase with a charge density wave order. Basically it's not really important what the order is. It's just that there is this insulating phase with the gap which is determined self-consistency by the distribution of the electrons. And then they add some dissipation. So they have the couple this system to an external bath which determines the temperature and they add dissipation. So they allow for the exchange of electrons between the system, the habit system and this external bath. And they find one that you need an electric field which is of the order of the insulating gap to collapse the insulator. So again, they find that you need some very large electric fields to destroy the insulating phase. And then they also find that the electrons are heated up. So to an effective temperature which is determined by the E squared. So the Joule heating term divided by the dissipation rate. So how efficiently you dissipate energy into the bath. And basically they see this by looking at the electronic distribution of omega's function of the energy omega. And they find that it can be fitted by a Fermi Dirac distribution with a renormalized temperature. So when you increase the voltage it's like you're increasing the effective temperature of your electrons. So they basically see that there is this heating effect of the electrons. And something similar happens in another model another work which is... So this model is a one-dimensional antiferromagnetic model with the Madeleine-Swedler transition. And again, the electrons are coupled to an external bath which is assumed to be at zero temperature. And there is a certain dissipation rate of energy between the electrons in the system and the thermodynamic bath. And again, what this... Put Han and collaborators find in this paper is that there is a renormalization of the electronic temperature for both the metallic phase and the insulating phase with the effective temperature which can be different between the two phases like the normalization effect can be different and stronger in the metal usually. But it's always related to the strength of the electric field. So the larger the field you apply to the system the larger the renormalized effective temperature is. And you also find again that you can collapse the insulating phase if the number of excited electrons due to Landau-Zinner tunneling is enough to destabilize the insulator. And what they do actually is also to study the phase diagram. So they study the behavior of the gap with the insulating phase delta as function of the applied electric field. And they see that like if you start from the insulating phase then you need to go up to a certain large field to collapse the insulating phase. So above this threshold there can be only a metallic phase. But then once you are in the metallic phase then you can decrease the field down to a much lower value. And at this point the metallic phase becomes unstable and you go back to the insulating phase. So they find that there is a large hysteresis cycle. And this is essentially due to the fact that the renormalization of the temperature is larger in the metallic phase. So when you apply the electric field to the metallic phase you have a larger dissipation. So like larger joule heating production so you have a larger increase of the effective temperature and basically you can stabilize this metallic phase to down to much lower temperature. And what they do is also use a phenomenological pseudo-free energy for the non-equilibrium phase which is here is just like a schematic picture. They, I don't think they actually have formula for this free energy. But like you can look at the minima of the free energy to find the values of the insulating gap determined by this free energy. So the values where the gap satisfies the self-consistent equations of the system. And then I just want to show you this other model which is part of the work I did during my PhD. And in this work we introduce a new mechanism which we can destabilize insulating phase in a correlated system. So we study one dimensional system with the charge density wave order. So there is an insulating phase with a certain gap delta which is determined self-consistently by how many electrons you have in conduction and balance band. It's like a BCS-like self-consistent equation. And we consider a sort of dynamic picture in which we have electric field which is changing the distribution of the electrons or the thermal electrons in the conduction balance band. Then we have, we consider several relaxation processes and the most important one is the relaxation process between the two bands, so the inter-band scattering. Now the main point on this work is that when this inter-band relaxation is as an untrivial structure in energy or in momentum. So when, for example, it's picked at K equals zero which corresponds to the smaller, the point of minimum gap, then it's possible to increase the number of carriers in conduction band by applying an electric field even without land-outs inter-tanneling. And the reason is that when you apply an electric field, you're increasing the energy of the electrons in conduction band. So you're essentially sweeping the electrons away from the minimum of the gap to higher energies in the conduction band. And these higher energies have a lesser future relaxation. So these electrons which live at higher energy where they should not be in equilibrium, they can survive longer because the relaxation rate here is smaller. So essentially you can have this effect in which you are increasing the number of electrons in conduction band. And this can lead with the stabilization of the insulating phase. And what we see is actually, and we actually see that in the phase diagram. So you can have, we plot the behavior of the insulating gap as function of temperature and for different applied electric fields. And we see that when we increase enough the electric fields, then there is no insulating phase above a certain temperature. So we need both a combination of base temperature, equilibrium temperature of the system and large fields to destroy this insulating phase. And also what we see is that there is a dual heating of the electrons. So if we look at the distribution of the electrons in conduction band, we find that it can be described by a Fermi-Dirac distribution with the renormalized temperature and effective temperature, which is T, the equilibrium temperature plus the dual heating term. So how efficient, so how much heat you generate by dual heating compared to how fast you can, the system can dissipate energy. And in our work, we also did something similar to what Han the collaborators did in their paper. So we looked at the sort of non-equilibrium pseudo-free energy. So what we did was essentially to start from the self-consistent equation for the gap delta and then just integrate this equation over delta so that we obtain a functional whose local minima give us, whose stationary points give us the solutions for the self-consistent equation. And in theory, there is no guarantee that looking at the, like making a comparison of the values of this pseudo-free energy indicate what space is stable and what is made stable and so on. But like this can be, can give us an idea of like where the system wants to be. So like whether the system wants to be in the incident phase or a metallic phase at different electric fields. And we find some results which are similar like this is equivalent to what's found in that other paper. So this was like more of an introduction to the QD models that I've been used to try to describe these transitions driven by current or voltage. But the main point is that from all these work as the takeaway point is that you need a large enough field to heat up the electrons enough so that the incident phase becomes unstable. So like it seems from all these models that the driving mechanism in this driven transition is actually heating of the system. So you want to have a large enough field to cause a large enough heating of the system and destabilize the insulator. But as I was mentioning, this does not really explain, does not seem to not explain the very small threshold field you have in calcium-reduced material. Now just to give a brief introduction. So calcium-reduced is a system with the many different phases. So you can see them here in the space diagram. But what we will focus on is the transition which happens at around 350 Kelvin from a paramagnetic insulating phase to a paramagnetic metallic phase. So we don't have to deal with any magnetic effects. We just have a transition from an insulator to a metal which can actually be described by a typical motor disorder behavior. So we have electrons, so like the relevant electrons near the Fermi energy. They occupy some half field bands. So like that's the perfect in, those are the perfect ingredients to have a motor-like behavior. So there is this transition from insulator to metal. And this transition is also associated to a change in the structure of the lattice. So there is a difference. This is a very complicated material. Like there are many, many atoms in the unit cell. So like it's going to get, if you look in detail, you can get arbitrary complicated. But the point is that there is a difference in the lattice parameters. And so in the parameters for the unit cells, for example, the length of the cell in the z-direction changes across the transition. This can actually be seen by in like very clean, very clean way in many experiments. There is, if you measure the lattice parameter of calcium glutamate, then you see that you can drive the transition with temperature and you see this clear jump in the lattice parameter, the transition temperature. Or you can also drive the transition using applying pressure to the material. So this is like a very clear indicator of where the transition is happening. As I was mentioning, there is this transition can also be driven by applying an electric field with the very low threshold field, which is like 40 volts per centimeter. So it's a very small field. And you can see that by doing the same measurement for the lattice parameter as functionally applied electric field. And you can see very clearly this jump in the parameter at this threshold field. And you also see that in the current versus voltage characteristics. So you go from a very, very low conductivity here in the insulating phase and then you jump at the threshold field and then you have the conductivity which is metallic indication or metallic behavior. So now the point is that from the experiments, it seems that the global heating of the system is small. So they calculated how much the system is heated up when you apply this much electric field, this much voltage. And they can only account for an increase of the temperature around 10 Kelvin. While there is these experiments are conducted at room temperature and the transition temperature is like 80 degrees. So there is an increase in, so to have a heating effect, you should have an increase in temperature of at least 50 or 60 degrees Kelvin, which does not seem to happen to be explained by Joe heating. Now, the last experiment I want to talk about and then I will go on and talk about the modeling of the system is this nano imaging experiment. So there was this group in Stony Brook that was able to basically take pictures of the system. So this is a sample of this material, cut some root and eight at different currents. And they can see in, they can resolve between the two different phases. So like using the fact that the structure of the lattice is different in the metallic and insulating phase, they are able to tell apart the insulating phase, which is represented in this gray, light gray color from the metallic phase, which is as this darker shade of gray color. So they are able to actually tell for every current they apply where the system is metallic and where the system is insulating. And what this is that when you start from zero current at room temperature, so you start from an insulator and then you start applying current and then you can see that the system starts turning metallic and minus V and plus V here are the points where you applied the electrode. So where you inject current into the system. And they see that increasing the current, so this, the interface between metallic phase and insulating phase grows. So like it expands across the system until when you go to larger currents, you, the entire sample becomes metallic. So they found basically a region of currents, a range of currents in which there is quick existence between the metallic and insulating phase in the system. And then they also saw this very surprising thing, which is that the metallic phase always nucleates out of the negative electrode. So they basically did the experiment in which the negative electrode is on the right side of the system. And then they see that the metallic phase comes out from this negative electrode when they apply certain current. And then they switch the podardis. So they put the negative electrode on the left side of the material. And so at this time the metallic phase came out from the left side. So always from the negative electrode. So this kind of surprising because the system somehow knows about the direction of the current. So there is something which depends on the direction of the current is kind of like rare in physical systems. And so also this effect tells us like that joule heating is very likely not the main effect here at play here, because joule heating depends on the current squared. So it does not care about the direction of the current while the system somehow knows where you are injecting the current or where you're taking it out of the system. So all these experiments just to try and think of this phenomenon in a different way. And the work I did with my advisor and the Millis of Columbia is reported in this paper here from last year. And so some of the main points here are that the presence of the interface is very likely important for the physics of the transition. Because there is a, you can see in this experiment there is a coexistence of faces. So the interface is likely to play a role in the transition. And also while the global joule heating while the global heating effect may be small, it is possible locally you have a larger effect. So in particular when you're injecting current in an inhomogeneous way, so like you have a bigger current density near the electrons where you're injecting current and it's both then there you have a larger heating. So you have a situation which is not homogeneous. So locally you may have regions where you have larger heating than the average sample. And then there is the issue of the polarity of the dependence on the direction of the current. And the main point is that this may is very likely related to thermoelectric effects since thermoelectric effects are actually dependent on linearly on the current. So it's possible that we can interpret what is going on here in this material as the consequence of some pelkier heating. And I will explain later what this means exactly. So to study this system, we basically brought down some macroscopic equations for the local temperature of the system. So we consider the temperature not as the system as a whole, but like locally. So we had a local equation. We also took into account the presence of an interface. So basically we consider a system which can have a, so the system as a length L with W and thickness H. And this system is resting on a substrate which basically acts as a heat reservoir. So it's like a bath at fixed temperature to zero. And then our system can have like existence of both metallic and insulating phases. So we have a metallic phase. So here there is just a sketch of a possible geometry which is a wedge-like geometry. So it's like the electron, one of the electrons will be here and there is a metallic phase nucleating out of this electron. And then there is an insulating phase and there is current flowing in the system. And what we do is to write the balance equation for the heat in the system so that we can determine the temperature of the electrons and the temperature of the lattice. So for the electrons we have that in a steady state we have zero equal to, then we have the joule heating which heats up the system. It's the resistivity of the electrons times the current squared. And then we have this Peltier heat term which is basically, which is minus the temperature T times the current density J times the gradient of the feedback coefficient. And then we have the diffusive term in which the temperature which there is diffusion of heat proportional to the conductive, thermal conductivity of the electrons. And then there is an exchange of energy between the electrons and the lattice. So basically the electrons can dissipate the energy produced by joule heating and these other effects into the lattice by exchanging energy with the lattice. And then we can write down the equation for the balance equation for the lattice. So we're always in a steady state we have zero equal to we have diffusion of energy of heat for the lattice. And then we have plus the energy which is flowing from the electrons into the lattice. And the important thing is that the lattice is the only one that can dissipate energy into the external environment. So through the substrate there can be diffusion of energy so that our system can lose heat by diffusing energy into the substrate at the interface at the bottom surface of the system. So we have we impose the boundary condition at the temperature of the lattice at z equal h. So at the bottom surface of the system is equal to zero. And basically what we can do is solve these equations and find the temperature of the electrons and of the lattice at any point in the system. Now, so we're saying this spelt effect, this term minus j times the gradient of the sieve coefficient is very good for us because one is linear in j and also depends on the direction of j. So it depends on the direction of the current. So it correctly gives the polarity effect with the experiment was telling us there is in a calcium route and it. And more importantly, when the current flows from the phase with the larger sieve coefficient to the phase with the lower sieve coefficient, there is generation of it at the interface. So this term here is positive. So you produce heat at the interface. So you are hitting the system. And it's actually consistent with calcium route it because in this material the sieve coefficient in the metallic phase is approximately zero. And then the insulating phase is large and positive. So it means that when the current flows from the insulator to the metal, then there is heating at the interface which corresponds to situation in which the metallic phase nucleates out of the negative electron is actually consistent with the pictures I was showing you earlier here which the metallic phase comes out of the negative electron. And then also there is this, the point that this effect is called K-heating term is proportional to the gradient of the sieve coefficient. And since the sieve coefficient has this big jump at the interface between the insulator and the metal will be a large generation of it at the interface which will compete with the diffusion and dissipation. And so concerning dissipation we can model it as just linearly in the temperature difference between the electron temperature and the lattice temperature. And the point is that this coefficient here which tells us how fast the exchange of energy happens between electrons and lattice is usually very large. So this exchange of energy is very fast which means that we can assume that the temperature of the electrons and of the lattice are at all times almost the same. So like the difference between the two temperatures here is much smaller than the temperature itself. So we can approximate so we can reduce our two temperatures model to just one temperature model for the system. Of course, this more like if we want to go more in detail this actually depends on the thickness of the system but the point is that for this material same the systems are usually thick enough for these two temperatures to be very close to each other. So we don't have to worry about a difference in temperature between electrons and lattice. And basically we can just write down one heat balance equation for just one temperature in which now the thermal conductivity is given by the sum of the two thermal conductivities of the system or the electrons and lattice and the boundary condition at the bottom surface is that the temperature is equal to two T zero to the bath temperature. And this essentially so this boundary condition tells us that this diffusion into the substrate into the environment is the only way for the system to dissipate heat the heat generated by Joule heating or Peltier heating. And in particular, very thick samples, very thick system dissipate less efficiently because the dissipation is proportional to a couple times the temperature divided by the thickness H of the system squared. So the fact like this calcium with the systems are usually larger than typical samples for other materials actually tells us that dissipation may be very inefficient in the systems. Now we can take this equation and for a given current injected into the system we can solve everything for the temperature and what we find. So we can get like some analytical insight for like very specific cases but maybe we'll skip this and just go to the numerical results for the system. So basically I solved these equations numerically for certain parameter values which are consistent with the experiments for this material for calcium and luteinate. And I found that there is, as you can see from these plots from this color map of the temperature there is a substantial increase of the temperature of the system and the temperature can even go above the equilibrium transition temperature so we can have a separation formation of the metallic phases near the negative electrode while the rest of the system stays insulating stays insulator and you see that in from the side view here in which you see the metallic phase that goes down to certain depth of the system and then from the top view in which you see also this semi-circular shape of the metallic phase coming out of the negative electrode and you can also observe like how the one deep profile of the temperature on the line joining the two electrodes here. So this shows that for certain parameters for a current which is for a value of the current which is consistent with the experiments you can actually induce this transition simply from the Peltier heating generated at the interface between the metallic phase and the insulating phase. And then I did a check in which I reduced the value of the C-beck coefficients by one order of magnitude and I saw that to see a transition so to nucleate some metallic phase I needed to increase the current but for this value of the C-beck coefficient the joule heating became way more important so the interface to the phase boundary looked way more symmetrical so there was also a metallic phase nucleating out of the positive electrode which means that in this case the C-beck coefficient is less important so joule heating which is symmetric in the direction of the current is the main mechanism of driving the transition so you will expect something which is very symmetric and then you can also look at a different geometry of the system so instead of like a square geometry I looked at the rectangular geometry and you can see how increasing the current leads to different shapes of the interface so you can see that the metallic phase the boundary between the metal insulator push it through halfway through the sample. Okay and then I just want to show one last experiment so this data were published shortly after we published our theoretical model and in this experiment it's like by a group in Kyoto University in Japan who were able to measure the local temperature so they did some infrared imaging of the calcium routinate under different currents and they were able to measure the local temperature of the system and you can see from this color map that there is this asymmetry between the electrode so you have like the negative electrode here and the positive electrode here in this picture for example and you can see how the negative electrode has a larger temperature so like it's hotter here while the positive electrode is slightly colder than the negative electrode and you can see also that in the plots on the right so basically they found that reversing the polarity of the current you always see that the negative electrode is slightly hotter than the positive one so basically so this data basically confirmed that Peltier heating is actually relevant in the system and you can see actually the difference in temperature between the two electrodes is linear in temperature it's linear in the temperature so showing that this Peltier heating is actually one of the main mechanism responsible for the transition in calcium routinate. Okay so in conclusion in this work I studied this metal insulator transition in this specific material in this routinate driven by an applied current and using some macroscopic equations for the local temperature I found that the interface between the metallic and insulating phase that can be a large Peltier heating due to the difference in the feedback coefficient between the two phases and this can lead to polarity dependent effects which can actually explain the driving of the transition in calcium routinate so we explained that this transition here is due to a heating caused by this Peltier effect and not to some purely electronic mechanism happening in the system with this I want to thank for your attention I will be happy to take any question. Are there any questions? So maybe I have a question or also Andre, Andre please, sorry. Oh, it's just a curiosity for me. I remember a few years ago there was a story about calcium routinate and the observation of the Higgs mode and subsequent decay of this mode. The story was that basically its ground state is spin orbit singlet with some temperature there was excitation of triplets which lead to this Higgs mode but there were other claims that the system is actually can be described as Higgs number 24 magnet. Are you talking about calcium routinate or? Yeah, calcium routinate. Because, okay, so here I am focusing on this higher temperature transition. So it's like the one at 350 Kelvin. So it's between two paramagnetic phases. I think you're referring to the transition. So there is a second transition between anti-pheromonetic phase and paramagnetic phase, lower temperature. Yes, I didn't study that. I'm not sure what you're asking for. I think it's a bit lower temperature. So yeah, I was just interested in how it's applied to this study. Yeah, so. Any relation I'm not forcing? So there have been some more recent experiments in which they applied current and lower temperature and they saw like a transition to a metallic behavior but I didn't study in detail the moderate dose temperature. In theory, the equations I wrote should describe the system of the lower temperature provided that I use the parameters for calcium routinate at that temperature. So I need the C-becoefficient, the value of the C-becoefficient resistivity and so on at this temperature of this transition. I didn't try to do that honestly. Okay, thanks. I have a question like is there a physical reason why to understand why this effective temperature is proportional to the applied field? You mean for the Peltier heating term or? So okay, maybe at the microscopic level. So the idea is, so here what in like this model, this picture here is for electrons in conduction band and when you apply an electric field you are basically heating up these electrons. So you are giving energy to these electrons and the energy you are giving is proportional to E squared to the Joule heating term more or less and then you are dissipating energy with some ray gamma epsilon for example. There is like some, there will be many processes with which you can dissipate energy. So it's basically a balance of like how much energy over time you are giving to the electrons through Joule heating and how fast you can dissipate the energy through several processes which depend on the microscopic model of the system. So basically the renormalization in temperature will be proportional to Joule heating divided by the rate of relaxation. I don't know if this answer your question completely. Yeah, I mean, I'm just confused because maybe we can discuss it later like, I wonder if it should be proportional to the E squared and why is it proportional to E? Because of the dissipation is basically proportional to E. Okay, no, sorry. So here in this model here it's proportional to E squared. Okay, I know, I think maybe go back one. Yes, yes, so you write here it's proportional to E. It's because they are using a phenonic bath basically. It's like the natural of the renormalization, like the dissipation is slightly different. And basically, okay, maybe I didn't show it here, but when they study this model, it's finite temperature. So they study this at zero temperature. But then if you also, if you take instead the phenonic bath to have a finite temperature, the formula for defective temperature is the square root of this quantity squared plus the temperature of the bath squared. So you have a correction. So you always have a correction which is proportional to the square of the electric field. But since in this case, you start from a phenonic bath at zero temperature, then they simplify and get this thing linear. But it depends on the absolute value of the electric field. In any case, okay. Okay, thanks. I think we have some questions. Yes, so you may have mentioned it during your talk, but in which case I missed it, but this difference of temperature between the anode and the cathode, it can be understood as far as I get it as a breaking of the particle-hole symmetry. However, my question is, where in your description does this breaking of the particle-hole symmetry occur? Was it explicitly in the microscopic model or was it, is it a spontaneous breaking? I might have missed that part. Yeah, no, that's a very good question. Yes, so like it's implicit in the structure of the system, the microscopic model. I didn't show it here, but you can see like essentially the C-back coefficient is a measure of particle-hole symmetry. So if you have a completely symmetric structure, then the C-back coefficient is zero. And this is actually what you have in the metallic phase where calcium glutamate is quite symmetric between particles and poles. While in the insulating phase, if you look at the energy structure and the density of states of the system near the Fermi energy, you have an asymmetry between particles and poles. And this leads to this C-back coefficient being different from zero. So essentially I would say that the breaking of symmetry is like hidden in the fact that this gradient of C-back coefficient is non-zero. So if you have no symmetric breaking, then S is zero, and now you have like the distance disappear. So you don't have this effect. Yes. Okay, but this cannot be seen by looking at only the dispersion relation and the feelings. Right? I mean, it didn't look to be the case in the few graphs that you show afterwards. No. You mean? So it's really a breaking in terms of the transport properties? Yes, I guess you could say that. Yeah, it's just like a symmetry breaking at the level of a microscopic structure in the insulating phase. It's like, I mean, you can look at, you can find that there is this asymmetry if you do like ab initio calculation for the energy structure. Or if you look at the expert, there are some experiments measuring the density of states of the system or the C-back coefficient, and you can see this asymmetry, but from like a more intuitive level, I don't know, but yeah. Okay, I see. Thank you. Any other questions? If not, let's thank the speaker. And and this I think brings the end to our session with this two very nice talk and also today. So I'll hand over the stage two today. And yeah. So what do we do from now? We... So thank you, Arithra, for sharing the session. So before closing the session and the day for this conference, I would like only to remind you some technical information for tomorrow. So tomorrow the conference will start at nine. If possible, it would be very nice for the speakers to try to connect 10 to 15 minutes before to try if everything is working for them. And also, as you may see, the program has been updated. And so we have confirmed that we have a photo session tomorrow after 12. So directly after the last talk of the third session tomorrow morning, we will have a photo session where both all the speakers and all the participants are welcome. Okay. So you will shall receive an email with the link to the Zoom meeting and not the Zoom webinar to connect at that time. But I would remind that to you tomorrow as well so that we can have a group photo. It doesn't replace the real face to face, but it's the best that we can do online. And that's about it with the technical information that I wanted to tell you. So I thank you again for your participation. And I hope to see you tomorrow for more interesting talks.