 Buongiorno a tutti, è un grande piacere di avere Liliana Rachea qui per la CMSP Colloquium. È un piacere bio sketch di Liliana, che ha la sua PhD in Università La Plata in Argentina. Dopo questo, si muoveva per postdocs, including a periodi di postdocs qui a CISA in Trieste. Ora, Liliana è professore all'Università San Martín in Buenos Aires, e si è stata appoggiata come professore di Numbolt, fra l'Università in Berlin, e si lavora con Max Planck. Liliana's work is in various areas of condensed matter, strongly correlated system, quantum transport and today she will talk about some topic in heat to work conversion in quantum regime. So please, thank you. Hello, thank you very much, Sara, and thank you for the introduction and also for the invitation. I really feel honored by this invitation. Although I have to say that I am not used to speak while being organizer of an activity, this is one of the few exceptions, but yeah, for this reason I am... Can someone... Yeah, yeah, here it goes. For this reason you will see that as there are several participants from our activity here, that I will make at several points of the talk, a sort of overview, and summary of those topics of our school and workshop which are related to the main topic of the talk, which is heat to work conversion in the quantum regime. So, yeah, as I said before, the context of this talk is in the middle of this activity, where we are having a series of lectures and talks on all these topics, which are very active topics nowadays. These are all listed there, quantum heat transport, quantum dissipation, quantum thermodynamics, quantum thermal machines, quantum thermoelectrics, quantum fluctuations, Maxwell-Demons, and Floquet engineering. And here are all the communities, at least I can identify four big communities, which are thinking and having ideas on all these topics. In several of them, several of them are closely related to the main topic of the colloquium. So, let me mention some, first some few examples, and then I'm going to go deeply to other cases which are closer to my area of research. The first thing that comes into our mind when we talk about heat to work conversion are these thermodynamic cycles that we all studied in textbooks in the school and also in general physics, in which we have some thermal machine operating, and also doing some mechanical work, and typically we have these two regimes where we invest work in order to extract heat from a cold reservoir and to inject it into the hot reservoir, or the other way around to use the heat flowing from the hot to the cold reservoir in order to generate work. And we can do it very slowly following these cycles, this is one of the famous cycles, but it turns out that these kind of problems are nowadays formulated in other very different systems. So here we see that we have, we talk about these reservoirs, these reservoirs are macroscopic systems, but not only the reservoirs, also the pistons, also the central pieces of the machine are also microscopic systems. But it turns out that nowadays people are doing mainly the experimentalists that are doing fantastic things with different fields, managed to do these kind of cycles with a single ion for instance. This is a slide from the lecture of Ferdinand Schmidt Keller from our school, and where he explained all the details to how to trap ions, just a single or a few ions, and to operate with optical means in order to produce something very close to this cycle I showed before. And there are plenty of questions in this case, so here instead of a piston, a macroscopic piston, we have a single ion, and instead of having a macroscopic reservoir, the reservoir, what plays the reservoir here are lasers, and so the devices are completely different and it's not so easy to translate the concepts, the usual concepts of thermodynamics to this context. And actually in addition to the lectures of Ferdinand, we had very animated discussions precisely today after the talk of Kuritski related precisely to this type of example. Another popular example that we know from general physics is this Maxwell demon, this Gedanken being invented by Maxwell, who is able to operate a small slide, a small window separating two gases in order to move to collect all the free particles in one side and the hot particles in the other side. And we also had a series of lectures and talks related on how to generate this scenario in small systems, and one example is this one, so this is from the slide of the lecture of Yucca Pecola, which is an experiment where this mechanism of the Maxwell demon is implemented in a small device, a small electronic device, so this is the typical size, and here electrons are confined and moved by means of applying the voltages, and something very similar to what happens in the case of the Maxwell demon can be implemented here. So there are still also questions on to what an extent the context is quantum system, so in the case of the atom of the ion, we have a single ion and a single ion is a quantum system, but there are also questions on the quantumness of the operational mode itself and this was addressed for instance today by Michele Campici where instead of using mechanical or classical mechanism to generate the cycle, one can rely on quantum measures, which is something which is purely quantum, measurement purely quantum mechanical measurement process in order to also exchange energy, I would say, between two systems in contact to hot and cold reservoirs. So these are the questions that are around, and then I move to another scenario, which is perhaps a clearer context where we have really heat to work conversion and this is the context of solid state physics of condensed matter where we have electrical heat to work a conversion. As an area, this is a label that receives the name of thermoelectricity and this is the thermoelectricity as explained in Wikipedia. So thermoelectricity as explained in Wikipedia is the possibility, is the effective conversion of a temperature difference into an electrical potential or vice versa and we can use it into operational modes, we can use the heat flow from the hot to a cold reservoir to generate an electrical current in an external circuit or, sorry, here, the heat from the hot to the cold to generate an electrical current or to use an electrical voltage in order to extract heat from a cold reservoir and in this way we refrigerate. So, formally speaking, it's quite easy to introduce the main ideas behind this. This was treated by Giuliano Benetti last week. We typically have the operational mode of this machine, it's the following, we have two reservoirs in which we apply a difference of temperature and a difference of chemical potential and in this way we have, this is a conductor here which enables charge flow along with heat flow. And of course here we have to obey some basic conservation laws. The first conservation law is the conservation of the charge so all the charge abandoning one reservoir should be injected in the other one and also a conservation of the energy. Given this and assuming that there is some linear response between the fluxes, so the fluxes, the relevant fluxes here are the heat fluxes and the electrical fluxes and the forces which generate the fluxes are named in thermodynamics affinities and these affinities are basically the difference of chemical potential and the difference of temperature so simply assuming that there is a linear relation between these two quantities and some basic constraints for these coefficients so these coefficients L receive the name of on-sagger coefficients for transport coefficients as these are linear so these coefficients depend on the equilibrium properties of the system and because of that they obey micro-reversibility which means that they obey this so-called on-sagger relation so given these constraints for these coefficients and given also the constraints that we have to obey the second law of thermodynamics this means that the net rate of entropy production in this system should be positive we find also some additional constructions for these coefficients and from there we can formulate some convenient expressions for the efficiency the efficiency is as any efficiency what we want to get divided what we have to invest in the case of a heat engine what we want to get is electrical power it is the numerator and in the denominator we have what we have to invest this is the heat flowing from the hot to the cold reservoir and in the case of a refrigerator is the other way around in both cases this efficiency is bounded by the Carnot efficiency and using the relations I mentioned in the previous slide we can very easily show that if we fix the difference of temperature this efficiency can be parameterized in this way we have here the Carnot efficiency and we have here some quantity which is a combination of the Disson-Sager coefficients and which receives the name of a figure of merit so this figure of merit is typically one of the quantities that people use to characterize the quality of the thermoelectrical device so this is a very nice review by Giuliano and Robert Whitney both the three of them participated here some of them are around where they precisely mentioned that in order to get a reasonable efficiency of the device we need this figure of merit to be at least three so as I didn't mention before that we can notice that the limit the ideal limit of the Carnot is achieved with this figure of merit goes to infinity so at least three gives some efficiency which is of the order of one third of Carnot efficiency so which is the status of the applications this is a new recent review of thermoelectricity and this is the status in the global context so we see here the power generated by different means we have here thermoionic and many other types of heat to work conversion and we have here temperature we see that the temperature goes up to very large numbers and we have here this figure of merit we can also appreciate that the less efficient process is precisely thermoelectricity any other are much better but still as it is a nice way to generate a very nice and clean way to generate electricity from waste heat for instance there is a strong interest to investigate and to find better and better materials which are more and more efficient and this is the status of the research so you see that as time evolves we have new materials which are pretty close to this ideal not ideal but acceptable number three this is a nice application in spite of the fact that it is not very efficient it is used because it is a clean mean it is very stable and it is used precisely in the space mission and this is precisely an example where the heat is produced by is produced by some not reaction so there is some isotope which is unstable and in this way heat is generated and this generated heat is precisely what is used to I didn't remember the name so the question is what happens if we go to so these examples are very macroscopic and let's come back to the subject of our school what happens when we go to the quantum regime so what is quantum so what do I mean by quantum what I mean by quantum are cases in which the electron propagates coherently so coherently means that propagates along some distances typical distances are some micrometers without experiencing an elastic scattering so as we know within the materials we don't have only the electrons the electrons have to live with the phonons and they always scatter with the phonons and this is the origin of the electrical resistance but as we go down to very low temperatures typically temperatures which are much smaller much lower than the one Kelvin these processes are very slow down so the phonons are not so active and if we work with devices which are actually of the typical size of the domestic devices that we have for instance in our computer we can achieve this regime where the electrons propagate coherently Importantly not any device has this nice property of thermal electricity in order to have thermal electricity we need this important ingredient that is the so called the particle hole symmetry breaking or the energy filtering this is because if we put one electron reservoir close to and we connect it to another electron reservoir at a lower temperature and if these reservoirs are of free electrons we have electrons which flow from this left to from the hot to the right to the cold part but as they are hotter we also have a counter flow of electrons that try to occupy the empty places left here and the net effect is that we have two reservoirs at different temperatures but we don't have a net a net particle flow a net charge flow so if we put some filter here we can somehow avoid some of this counter flow and in this way we can get a thermal electricity so the device must have this important ingredient and interestingly in the coherent transport regime all the information about this energy filtering is encoded in some function here which has the name of the transmission function so you see here I have the relation between the fluxes the heat and the charge flux and the affinities we have here these else are the transport coefficients and these transport coefficients in the coherent regime are expressed in this simple way so this is the derivative of the Fermi function this is a simple integral this is the temperature and all the information of the device is encoded here so we have to engineer the device in order to have some appropriate transmission function in the coherent regime in order to achieve some good thermal electric performance so these are the typical names that are given to the transport coefficients the diagonal ones are related to the electrical conductance and the thermal conductance and the off diagonal ones in this I am talking about this matrix here this L hat is a matrix so the off diagonal ones are precisely the ones which produce the thermal electricity and receive the name of Peltier and Sivec coefficients some fundamental bounds that these coefficients obey in the coherent transport are the following the maximum conductance that we can get for a channel for a given channel corresponds to this quantum of thermal conductance this is the electron charge so this is a quantity that depends on a fundamental quantities and the same something similar happens with the thermal conductance except for the fact that we have here T the temperature but all the other coefficients in front of the temperature are a fundamental constants in many cases but not always it is obeyed this so called Wiedermann Franz law and importantly one of the one can show that in order to achieve a good thermal electric response or thermal electric response we have to somehow also violate this law so this is usually the case of the quantum dots the quantum dots were explained how to construct them by the lectures of Lawrence Molenkamp they are typically constructed in sandwiches of these semiconducting structures those semiconducting structures which host there two dimensional electron gas in order to move to this pointer some two dimensional electron gas that the experimentalist managed to operate with gates so these gates are built with nanolithography so this is a device that typically has the size of one micron and with these gates they can make operations so they can find in this small part the electrons here this small part behaves like a sort of atom so with discrete levels which are operated by these gates and are also the currents generated so this plays the role of the reservoir so this leads here play the role of the reservoir I showed in the sketch before these quantum dots devices are precisely the typical devices that show thermoelectricity why because they have as I said before a set of discrete levels these discrete levels play very well the role of energy filters in this way enable the thermoelectric mechanism so this is a I borrowed this from the talk of Heinerlinke today or yesterday I don't remember already yesterday and this is another case another quantum dot also from this from the talk of Clemens were in addition of these discrete levels here they play with the electrons the effect of the electron-electron interactions and the effect of the electron-electron interactions is precisely a way to generate the so-called condo effect this is a strongly correlated state in which violates the Biedermann-Franc's law I mentioned before so this is some theoretical and experimental references and I also would like to mention this one by Misha here with the related idea I think that also the talk by Fabio Tadei is related to this idea but there are other systems which are also very interesting which contain quantum dots but they are slightly different and these are the edge states of topological insulators and these are states which are paradigmatic regarding the quantum coherence so the first example of the topological insulator is the quantum hole effect which was discovered by von Klitzing and several novel prices were related precisely to the fascinating physical features of this phenomenon it is a state that we obtain when we place the same quantum dot in a strong magnetic field in the presence of a strong magnetic field we know that even in the usual non-quantum case we have some cross voltage here the so called hole voltage but the peculiarity of the quantum regime and this is an effect that takes place at very low temperatures typically 300 millikelvin or lower is that for the hole resistance so the hole resistance is associated to the hole voltage the hole resistance instead of following this straight line presents these steps and these steps are very precise at some specific positions that are identified with the integer numbers or some special fractions the integer are the ones referred to, named the integer quantum hole effect and the fractional, the fractions correspond to the fractional quantum hole effect so these steps are so precise at some numbers precise numbers of this unit that this is used in metrology to define the unit of the electrical resistance and this can be understood so the understanding of this phenomenon is quite easy for the integer case is quite easy quantum mechanical problem we just have to solve the Hamiltonian of two dimensional electron gas in the presence of a strong magnetic field and we will find that the structure of levels for this system has this lambda level structure but as the system has boundaries as this is a topological state this system has also edge states and these edge states are very peculiar and very nice so the electrons in these edge states propagate chirally and because of that they are strongly robust and protected because there is no way to find some impurity and to back scattering and to move in the other direction and the fractional quantum hole is a much more complicated problem because here also play the role the many body interactions so the electron-electron interactions in addition to the strong magnetic field and the result is quite complicated but also very interesting state precisely in the case of these fractions that received the name of Laughlin state this was solved and understood by Laughlin mainly and he got the Nobel Prize because of that and this state also has edge states but here as a result of the strong electron-electron interactions the electrons live in a collective state so they don't behave they behave like a collective quasi-particles which don't obey a fermionic neither a fermionic nor a bosonic statistics they obey a so-called anionic statistics because of that this state is also very interesting in relation to the possibility of implementing topological quantum computing and it also has a host edge states in which the propagation is chiral but the quantities that propagates of the description here is a chiral propagation of bosons rather than fermions ok, we have very nice talk by Christian Gladley who is an expert on the quantum hall he is an experimentalist and he showed that with these edge states and producing here constructing here constructions it is possible to mimic to construct systems which mimic optical interferometers so they are really very interesting systems and he showed some recent results precisely verifying the fact that in this fractional feelings the quasi-particles behave as if having some fractional charge so this one-third for instance on two or two five so electrical transport in quantum hall has been studied for some years now since it was discovered there are still plenty of interesting things to unveil but thermal transport is rather new so to measure thermal transport in these small devices is quite challenging Nevertheless, in the recent time some time ago it started to appear some experiments in this direction this is one that I will say the first example where it was shown that not only the electrons not only the charge but also the heat propagates chirally here and in this this is a very nice experiment where it was proved that in one of these constructions of the quantum hole of the integer quantum hole effect it is possible to measure exactly this quantum of thermal conductance I mentioned at the beginning and this was recently also repeated in the fractional case This is another experiment devoted to study in some other fractional case heat transport so far only heat not no thermal electricity but actually so far no experiments at all on thermal electricity in the quantum hole but there are several theoretical proposals that are listed here these are more or less recent papers several of them in the integer but some of them in the fractional as well so let me very briefly tell you our work which was published last year where we showed that the charge fractionalization can be used to improve the thermal electric performance precisely by records of this thanks to this the violation of the Biedermann-Franc's law I mentioned before so this is the device so this is a gedanken experiment this is the device that we consider we have two fractions fractional states here and are connected by records to constructions through quantum point contacts through a quantum dot so we have this quantum dot which makes the place a role of a filter so this can also be constructed in this quantum hole structures and the idea here is that we have these electrons living in these collective states on this side also on this side but when they jump to the quantum dot they have to convert from this collective state to a true electron and this is the process this conversion is precisely the process which helps to break this Biedermann-Franc's law and can be used for thermal electricity this is just to show you that we solve equations I am not going to explain to you how we solve them but we write Hamiltonians for each of these parts and we solve them with the Keldish non-equilibrium green function techniques I am not going to provide any details about that I just tell you the result so this as I said before the process that generates thermal electricity is the breaking of the Biedermann-Franc's law unlike the picture I show at the beginning that holds for single electron systems where you have this filter and you have these electrons that leave some holes simply in this case when some of these some electron jumps into the quantum dot has to collect all the fractional quasi particles that are associated to it and in doing so it generates a sort of tsunami at the level of the energy and in this way the Biedermann-Franc's law is broken and this can be used for thermal electricity and the nice thing is that we can show and these are just some plots this is the Siebert coefficient that this fractionalization helps to enhance thermal electricity and this is the result for the integer and these are the results for the fractional and we see that this Siebert coefficient increases and so is the case of the figure of merit so these are some estimates taking into account on this process there is also room to investigate thermal electricity in a quantum spin hole so in other topological insulators an example is this quantum spin hole nanostructures they were this is the first paper in which this material was shown that to be in the topological state with the peculiarity of hosting now H states which are not only chiral but they now also have spin so it is interesting because one of the H states are electrons which propagate in one direction with the given spin orientation and the other direction with the opposite spin orientation and this was also the subject of some talks here in our activity again there are some some theoretical proposals to generate thermal electricity in these setups in different configurations with the constrictions that generate interference to have some cross effects applying here difference of temperature different of potential to consider non linear effects so there are plenty of things they are also quite creative because you can also play with the role of spin where you have you can put in addition considering in addition not only an energy filter also some magnetic field it is possible to filter the spin and to for instance cool one side rather than more than the other and so on there is room to very creative devices and operational modes but I would like to very briefly this setup which is really very simple and it is the following we have the H states here in contact to magnets and we applied at these parts the difference of chemical potential the difference of temperature and this is the Hamiltonian so for those who are from high energy physics can read here the one dimensional Dirac Hamiltonian with the mass there and this was also mentioned in our school there is this already old proposal where it was shown that the optimal thermal electricity which would correspond to this infinite figure of merit could be achieved when the transmission function has exactly the behavior of a delta function or something closer to it so it's closer it's close to a Carnot but it doesn't achieve a Carnot and there is also this paper by Robert Whitney here somewhere where he showed that in order to get maximum power maximum electrical power from these devices you need some transmission function which has this shape the shape of a delta of a heavy side step function and this is the bound for the maximum power you can extract for a given channel and you achieve this when you get this form for the transmission function and this is the transmission function that we get in this simple setup so we see that we have here because the mass term because the coupling to the magnetic moments opens a gap in the spectrum we have here something which has an envelope a heavy side function of course in addition to other oscillations but it's important that the envelope is this heavy side function and because of that we get as a result some power which is very close to the optimal bound which can be achieved and Daniel is presented already a poster almost I think more interesting is the following that if instead of considering just one magnetic moment we consider two magnetic moments with different orientations like this configuration we have a realization of the so-called Jackie Revy model in which corresponds to Dirac Hamiltonian with mass having a change of sign in space as we move we have some mass term which changes sign here in the magnetic domain and this model is known to have a zero mode which has also a topological origin so we have here a states of a topological insulator this device as a whole also hosts some zero mode which is topological in origin and this is very this zero mode is strongly localized and because of that produces this delta function like peak which of course is not exactly a delta function because the magnetic moment has some finite length and it has some finite width and this is very good to get a very high figure of a magnet so this is the results in two minutes I just mentioned the possibility of introducing a dynamical effects the first slide I showed was a sterling machine where we had in addition this time dependent this periodic device to play with the work and now the way to generalize the thermoelectric treatment in linear response in order to include this dynamical effect is by means of doing some linear response adiabatic expansion which this is our work but it's very close to I mean it's basically a generalization of what Juan explained in the lectures and there are other ways of looking at it for instance by Anatoly Polkovnikov and also Chanin in her talks also there is some work by Michele Kampisi and Peter Henghi and I showed you the sketch of the formalism in a while but the idea here is that we have reservoirs where you can put difference of chemical potential difference of temperature but in addition you have the possibility of exchange work with the ascent which makes this cyclic perturbation and the way is precisely this to consider the evolution operator in quantum mechanics treated in this way as a frozen contribution plus some correction this is what we call the adiabatic correction which is proportional to the rate of change of the time dependent parameters of the Hamiltonian so the name adiabatic is because we are assuming that the rate of change is very slow this velocity is very slow and because of that we do a linear expansion in this velocity well, we are using right now just to mention this framework to analyze these thermal machines from a geometric perspective this is a work which was presented in posters by Vivec and Pablo who are around and it is quite interesting because we can introduce the geometrical concept so this adiabatic movement can be expressed in this geometrical path in the parameter space which allows a characterization in terms of metrics and carbatures and so on which is I think quite interesting so this is all I wanted to tell you just I would like to acknowledge and thank the collaborations with my PhD students Daniel and Pablo are around Lionel is in Buenos Aires Mariano is an experimentalist so we are also doing some experimental work in relation to the quantum Bay Hall effect and we are also collaborating with the PISA group and the students there Gianmichele Blasi and Vivec who is also around and also ex students and postdoc Florencia Ludovico was also around in blue are all the experimentalists there are some in Buenos Aires and we are also collaborating with the people from colleagues from Max Planck Stuttgart and PTH and the other colleagues from Bariloche, Argentina Eduardo Fradkin, David who was around, Rosa who was last week Misha Moscales from Ukraine Alessandro Fabio around Matteo from PISA, Saro and Felix von Oppen from Berlin so thank you very much Thank you, so questions so I can start with a question so about the topological zero moment it's not really important it's topological do you really want to have just about say the zero energy right or not? I think it's important because if it is topological you can argue that it's robust against some perturbations and I think this is what this So you want it to be stick that at the Fermi energy at the zero energy no no you don't so if you have other orientations of the magnetic moment you can move it from the zero energy but yes Ok With your suggestion about speak all device would non linear effects in terraria acousticity change towards you exceed up about or other way around how would it work? Well, non linear may be complicated to treat because maybe so here we are assuming that we are having a very weak contact between the two edge states but if you move away from the non linear you put a strong difference voltage difference for instance you can move to some other fixed point where your two edges get in touch so all the behavior can change drastically so here I could eventually play and I think that including non linearities but within still small voltage differences it could eventually improve With both but any of them if you increase it a lot you can move to another fixed point and your edge state start to behave completely differently So a general question in the field you showed the enhancement for the thermoelectric effect as the ZT parameter the question is are those results all at low cryogenic temperatures or are there any at the No no, these are all at low cryogenic temperatures Then the question is how do you feel about finding where the field is going and the possibility of finding this at higher temperatures Well, all these effects so it's impossible to get a quantum hole so these are these are basic this is basic research I'm not thinking on these two for applications Perhaps the only application that you can think about this is precisely in lab machine in lab that you are so it's very challenging to go to very low temperatures in lab so if in your device you have something that thermoelectrically helps to cool it's something interesting but of course you don't even have a quantum hole at high temperatures in order to get the effect you must go below one Kelvin so all this is cryogenic in addition when you introduce the effect of temperature the phonons as it was said many times during our activity phonons start to play an important role and plenty of these nice features are destroyed Let's start talking about Maxwell-Demond which is the fact that if you observe microscopic state then you can extract some work now this is in the no no but it wasn't it's not the Maxwell-Demond was mainly I mean if you make a measurement then this allows you to I mean like in classical example in the Zillard engine so what happens in the quantum domain does quantum in the quantum domain can you violate these classical relations or well so I'm not the expert in this field perhaps some of my colleagues can answer better than me but I roughly speaking I said that all these experiments and all these proposals are basically classical so the context is quantum because you are in this experiment by Pecola for instance so today we had a very long discussion about the quantumness of all these effects which I think is very interesting but it was at the very beginning sorry this experiment for example you have a quantum context but the process itself is quite classical so the electrons here you have electrons which are trapped one by one but they also are moved one by one so it's at the end you don't have any quantum effects playing a role in the demon mechanism itself we also had another talk today with the conclusion so I don't know if there are realizations of this Maxwell demon which are really quantum so thanks a lot Juliana