 Okay good morning. Well I want to start with thinking organizers for putting me in this program and it's obviously a great honor both because of the occasion for the celebration but also as we just heard also because of the institution ICTP where the celebration takes place. In 78 I was still in college I I just you know bumped into this book on telling phenomena insolence. As a matter of fact I actually bought a officially officially pirated copy in Russian at that time Russia wasn't a member of the copyright convention so it was like two rubles or something like that and it's actually quite amazing collection of papers. The table of contents was reading like who is who in condensed matter physics. Actually some people are here and it's one of the these were lectures given in 67 at NETA Advanced Study Institute and one of the organizers of the school was Stig Lunkvist who also published this collection and as it happens 67 was a year when Lunkvist started to the program of condensed matter physics at ICTP and since then he really put his vigor into creating a very vivid program and people who came to ICTP once kept coming and it's a really great place to be and this conference is obviously a point in case. One of the papers there kind of captured my imagination it was about telling anomalies and that's the first time I've heard about zero by some anomalies. It was experimental paper with some at that time unclear zero bias peak in telling conductance but the thing is that these lectures were given in 67. I got this book accidentally in 78 and it was kind of too late because in a year some smart and nimble people kind of scooped the cream of the problem but my affection for Fermi singularity physics and the zero bias anomalies kind of remains since then and every time there are some excuse for it so now this excuse is Marana states which are inherently zero energy states so that's basically brings me to the subject of my talk about Marana states which possibly can be formed in a superconductor which is doped by magnetic impurities. So this a few papers that we wrote together with Felix Van Oppen and his students young Peng and Falco Pianca. So here is the outline I'll tell a little bit about motivation that is a prediction of Marana states in very simple contraption and I will not tell much about observations I will not overlap with the next talk but I'll tell a little bit about some other implementations of the suggested systems for engineered Maranas and for that I will need to discuss in more detail the so-called Shiba states the electronic states formed by a single magnetic impurity in superconductor then I proceed to discussing chains of Shiba states and that will lead us to the notion of P-wave superconductivity along the chain and Marana states so that's basically the plan. So the subject of Marana states in commencement are flared up when it was realized that one may try to make a system which is made of more or less conventional materials and sort of engineer this exotic quantum states. So in these two papers what was suggested is to take a one-dimensional quantum wire this spinorbit interaction and spinorbit term kind of displaces the parabolic dispersion relations for the electrons so that the two species two spin species have different minima in momentum space and then put such a wire in proximity with superconductor and in these papers there was a simplest version of proximity effect where one thinks about only pairs being able to tunnel into superconductor and back but not single electrons so kind of single electron process were already taken out integrated out and in effect the Hamiltonian was written as a pair tunneling. So this description is good actually for energies well below the gap in the host superconductor and the proximity term would induce gaps around the Fermi level at all the momentum. Now if we in addition include magnetic field acting along some other direction so it doesn't commute with the spinorbit term then it's homogeneous field so it affects only k equals 0 states and it competes with superconductivity and upon increasing b it closes the gap at p equals 0 and then gap reopens but actually this opening corresponds to creation of a p-wave superconductor where basically spins are almost aligned along the same direction because of this term but still there are some spinorbit that skews them and that allows proximity effect and some induced conductivity. Turns out it's p-wave and one may expect some exotic states at the interface between p and s-wave superconductors. So that was this initial idea in this sequence of works and then it was realized that one can do away with the spinorbit and for that one needs a helical magnetic field instead of just homogeneous field. So if one takes just conventional p squared over m dispersion relation no spinorbit and imposes a rotating in space magnetic field then by a simple unit transformation basically a gauge transformation one can exclude the rotation of the field and map this Hamiltonian onto the one that I have just shown with a homogeneous field in this new rotating frame and some term that looks like a spinorbit interaction term. So kh here is a pitch of the helix created by magnetic field. So again the result is that one forms a 1d p-wave superconductor at the ends of such a 1d system because of the non-trivial topological structure of p-wave superconductor there must be zero energy states Marana states and that was realized by a number of groups by Daniel Los Carlobinacus and Carson Flansberg group in Kappelhagen. So next level in this saga was to towards implementation was taken by Princeton group collaboration between Bernabek and Alias Dennis groups and basically the idea was to replace external field by Zeeman, the external field that couples by Zeeman energy replace it with these magnetic ions which would couple by exchange interaction to the ethylene electrons. So basically this b here is magnetic moment of an ion number n and the rest is exactly like in the previous discussion. So basically they had kind of tight binding model with nearest hubs modeling the bent electrons, some proximity and magnetic field, effective magnetic field induced by ions, magnetic moments. So again one can just take what everything is known for wires and conclude that there must be Marana states. So that was kind of a step towards experimental realization and then indeed experiment came from the same group in Princeton. So turns out actually that ions, iron atoms that were embedded into lead like to order themselves ferromagnetically and actually the distance between them is quite small. It's almost at the level of the interatomic distance in lead. So it's ferromagnetically not helical but it's not a big deal because actually lead is a heavy element so there is strong spin orbit and the surface breaks the inversion symmetry. So spin orbit indeed produces something like a large bi-term. So in this respect the combination of spin orbit and the polarization, ferromagnetic polarization works like roughly like a wire with spin orbit interaction placed in a homogenous magnetic field. So one expects Maranas and again you may think that this deep in the density of states around zero which corresponds to the spectrum taken somewhere in the middle of the wire is a p-wave gap okay it's not a very well pronounced but still and if spectrum is taken at the edge of the wire at the end then there is some bump here which is zero bias anomaly and hopefully it can be associated with a Marana state. Now these are color plots for conductance at different biases as a function of position along the wire and as you see at zero bias indeed the conductance flares up just at the end of the wire of this chain of items just at the end and it's good and bad because on one hand it's a localized state but on the other hand it's actually too localized because the localization is just few atomic spacings and the coherence lengths in lead is much larger about hundred times larger so it's not clear how these two things are compatible that the gap is narrow but the state is strongly localized okay so that seems strange and it caused some quite strong statements at first and more cautious statements so essentially what I want to say is that without making any conclusion whether the observation does indicate or does that indicate existence of Marana states I want to tell that these two things are fully compatible and for that basically we had to to develop a bit more the model and to abandon this wire thinking where you think only about the pair stumbling into superconductor and consider the formation of the electronic states along the chain of atoms in a more on a more microscopic basis and that brings me to this use of erosin of states that are formed by magnetic ions so maybe I'll briefly tell about the model that we're using so it's basically it's Anderson impurity model in which the atomic shell of a magnetic ion is partially filled but cool repulsion prohibits putting more electrons on the shell when the atom is embedded into metallic host so if characterization is infinitely small then we still have a discrete level for additional electron above the Fermi level and the whole level below the Fermi level now if one puts in a finite hybridization then the levels of course broaden up there some resonant with a proportional to the coupling squared and what I just did he just wrote the mean field equations for the monetization of the of the shell of the magnetic atom and looked at the non-trivial solutions of the mean field equations and the atom in the mean field approximation retains its monetization as long as the width is small enough compared to the repulsion and and the level is not too far away so that this one goes goes up or the or it's not too close to the Fermi level okay so basically this region of the positions of the levels and of their widths is good for retaining magnetic properties of an ion hybridized with a non-magnetic host so I assume that we are somewhere here in safe region where gamma the level width is much less than you and just for convenience for pedagogical purposes I'll assume that the the energy ED is is positive so this this level is further away from the Fermi level rather than this one from the Fermi level so but it doesn't matter much sorry this one is closer to Fermi level this one but this doesn't matter much it's just for convenience so this is the basic models that we are using now let's look at some unrealistic case when the level width is very narrow and it will be to start with even more narrow than the gap in superconductor at the same time one more unrealistic assumption position of the level I will take so close to the Fermi level that this distance is less than the gap in superconductor okay so still we have without superconductivity we have a broadened particle level and broadened Fermi and broadened whole level now we introduce superconductivity and that means that there are no available states in the bulk in some strip of energies equal to delta and therefore the atomic level has no bulk states to hybridize these so it used to be broadened without superconductivity but due to superconductivity the broadening will go away and that will be a discreet level okay so basically one thing the specific activity does is that this atomic level remains discreet if it's very close to the Fermi level but now one more step suppose I have some magic wand and I can increase the energy ed so this level is pushed up and becomes a resonance and the trivial thing is that the discreet level still remains in the gap and it's so-called shipper state it's bound state inside the gap and it's well below the resonance formed by the bare atomic level now there are various ways to explain why it happens I will do it in very sloppy way just alluding to the level repulsion so in superconduct is in a superconductor the density of states diverges at the gap so there is sort of a level can you can imagine there is a level at delta or there are lots of levels is energy is close to delta and now when we put in an atom with a discreet level there is avoided crossing between the atomic level and this divergent at delta set of levels in the superconductor and that basically brings about a discrete level split off the position of the singularity in the density of states so there is a bump in the density of states at the position of the bare atomic level but in addition to it there is a discreet level due to the repulsion which is split off due to the hybridization and the splitting is proportional to some high power of hybridization parameter actually it's power eight right so gamma is t squared so it's power eight now if you look at the composition of the of the wave function as it corresponds to this local stage it contains of course some contribution from the d level but it's a it's a miniscule contribution if this level is high and mostly this shipper state is composed from the quasi particle states it's a superposition of the particle states and therefore it's slowly decaying in space so the shipper state decays in spaces one over r up to some large distances controlled by superconductivity so now if I increase the position of the of the atomic level then as I said shipper level goes up the wave function here when the level is closer to familiar is almost all in on the atomic level but when I go away the contribution of the d level goes down as I already said and this picture remains even if gamma is large so if I make a more realistic assumption about large gamma over delta turns out that this current becomes more shallow but essentially the same it just deteriorates at higher values of a d plus you and what's important that the contribution of d level becomes always small even if the level is close to the Fermi level still the wave function is mostly of the shipper state is mostly due to the quasi particles rather than the d level of the of the atom so one may have this mental picture that electrons have very easy time to hop away from the d level to the superconductor okay they cannot go away at an infinite distance because of the gap but the larger the ratio gamma over delta the more they are delocalized okay so basically there is a kind of a cloud of electron that is formed around the atom but this leaves mostly outside outside the atom at energies below the gap at Shiba state energies now technically if I look at this limit at you Shiba Rosinov case then basically what one needs to do is to solve Schrodinger equation with delta function potential the little difference from usual textbook or quantum mechanics is that the wave function here is a is a column of four numbers two four spin-off spin down and two for the particle whole components or so-called number space and one can solve this equation to get the Shiba states as a function of the control parameter which is exchange interaction times density of states and one can see that by changing gamma one can tune the position of the Shiba state from delta through zero to minus delta so there is some quantum phase transition when the level crosses the chemical potential now the wave functions corresponding to the to the Shiba state okay the states are doubled because of the structure of BDG equations but basically the spinners it spin up for say plus and spin down for say minus and these two components correspond to particles and holes now basically this solution that was developed in this seminal papers that I mentioned she used Shiba and Rosinov independently can be generalized for the case of a chain and basically this method is long known from nuclear physics in 50s what and it described in these two lines so basically you pretend to solve the equation by free transform then do inverse free transform and form out of the differential equation a form a system of equations which is foreign by foreign matrix with the kernels it depends on the spectrum parameter e and the properties of this matrix basically define the entire spectrum including the continuum and the Shiba states now suppose we tune the Shiba states close to the Fermi level so I press this gamma to zero to two ones worry then the Shiba states are well separate from the continuum and what we can do if you look at the at the set of states with energies well below the gap we can project the entire hyperspace onto the linear combination of the Shiba states so now we have to adjust the Shiba states for each atom and we assume that atoms have different polarization say they are in helix and suppose there is no spin orbit in the host okay so let's look at simple situation when there is no spin orbit in the host but the atoms are indeed helically arranged in terms of spins so then basically the axis of quantization is is individual for each of the spinners for each for each of the sites and one has to replace the wave functions with the corresponding wave functions which are adjusted to the proper axis of quantization so now out of this set of states we can form linear combination project this huge matrix onto the space and that leads basically to a tight binding Hamiltonian for electrons moving along the chain so one important thing about the tight binding Hamiltonian is that there are long range hopes because as I said the Shiba states are very slowly decaying even a distance is less than psi the decay is 1 over r so here is this tight binding equation again it's two by two matrix these are hoping matrix elements and these are pairings or induced superconductivity and matrix elements and as you see both decay hopping and pairing decay is one over x and now your eyes attracted probably to this sign that manifests the p-wave nature of superconductivity so first of all you can see that there are no there is no pairing for the same site if I equals J sign is 0 and if I if I permute I and J the induced gap changes side so now it's easy to understand where it comes from because basically the on each site there is a place only for electron with a given speed so say for this guy the spin should be up and for this one it should look somewhat sidewise now we take a pair from a superconductor Cooper pair which is in this wave superconductivity in the singlet and try to put such a state on to available space available states and the problem will be that this state will be orthogonal to this one if there is no some finite angle with the axis so this projection is not zero only if there is some pitch in the helix of the formed by the magnetic ions so that basically is the origin of this sun page x ij a function and it's clear because of the structure of the spin one-half wave function that sign should change the ways they this function shows so let's maybe try to do some simple estimates and in the rest I will I will talk about helical arrangement but as we discussed in the very beginning there is a mapping between the helical arrangement of atoms and ferromagnetic arrangement of atoms with spin orbit in the bulk so I will change I will interchangeably use the two languages and the translation is here so basically the product of the pitch vector times the distance between magnetic atoms should be thought of as some effective spin orbit angle and this is the basic translation between the two systems but for languages between the two systems so now let's try to make some estimates so first of all out of this matrix elements which correspond to low-bridge hopings let's look just at the nearest neighbors hopping so I replace this x by the distance between magnetic atoms a and I'll get the typical bandwidths for the hubs it's delta divided by kfa so now if I have the bandwidths and I know the brilliant vector which is given by 1 over a I can estimate the typical velocity in the band and that will be delta over kf right because a cancels out so this characteristic velocity now let's look at the inducing conductivity I'll do the same kind of brute force crude estimates replace x by a and as a result I'll get the induced p-wave gap as delta divided by kfa times the sine of the spin orbit angle ok so now we're almost done because if I look at the characteristic correlation lengths in the induced superconductor along the chain I have to divide the effective firm velocity by the induced gap and as you see delta cancels out so what we get as a result is a distance between magnetic atoms divided by the spin orbit angle if it's small then we can just forget about sine and rest right alpha so basically the key point here is that the effective velocity for motion along the magnetic chain is not the firm velocity in the bulk and it comes from this combination of tunneling events from magnetic atom into the conduction band of a superconductor and moving of the shiba state along the chain so that's basically all about the estimates now one can do a more detailed theory of course because after all it's a can be solved they can be solved by fair transform and one can get the full phase diagram in terms of the position of the shiba level of individual impurity and from a vector and depending on the ratio of the pitch vector and kf there's no topological state here because there are more than one crossings for each direction of k in this case or if the kh made smaller then there may be a single crossing and that gives rise to the topological state so there actually two topological states depending on the particular position of the shiba level above or below the Fermi below the Fermi level these both have non-trivial index and again I will not develop calculation but we just can calculate by standard means the top index and is plus or minus one for these two parts of the topological states and it's zero for this part okay so now what about my runners uh if we go through the transition again as I said there is no marina state in the in the trivial topological state and there is a there is some point of phase transition when the border is crossed the transition basically comes as the vanishing of this central part of the bend structure and forming of the bend structure of this type and uh in topological phase there is a bound state at the edge of the chain the wetlands now the happen the hubs are long-range therefore it's made to expect strong localization of marina states on the theory ground and indeed what we find analytically is the decay of the wave function at large distances is one over x log squared so it's a very slow decay and the scale here is given by the detuning from the phase transition point and it looks relatively terrible but it turns out actually that it's a very distant asymptote and if one looks numerically at solution at all distances this asymptote comes only after a rapid decay of the wave function with the skill that I just introduced a a over sin alpha s o over several decades so basically if there is small parameter which is pinot rectangle then there is a number of atoms in the chain over which the wave function decays exponentially with this rate so there are localized states along the such a chain now the drawback of the model that I discussed so far is that it needs fine tuning I started with telling that we think of shipper states find it tuned to the Fermi level and there is no reason to expect it because we're talking about basically atomic energies right so the embedded atom has typical level level spacings of electron volts and the the width of the level is also about electron volt and somewhere there okay and then we want to tune shipper level to the Fermi level so doesn't seem very realistic and what I think saves the situation in experiment actually iron atoms are closely spaced and that means that instead of you know separate under some impurities there is a chain and there are hops between the D levels of separate atoms and it forms a band and the band may be more wide than the distance between the level and the Fermi level then the bottom of the band will be filled by electrons and as I said it does require such a fine tuning so now the tuning is in EV range rather than MEV right so the only thing that we need is that the bandwidth is order of EV now on top of that all the atoms are magnetic but as I said also this is in terms of it's morally the same as having no spin orbit interaction as having spin orbit interaction and it's the same as having no spin interaction sorry and and skewed positions skewed orientations of the magnetic moments of irons so once again in real material it's lead and it's strong spin orbit and irons are far magnetic we may do the unity transformation get rid of spin orbit interaction at the expense of twisting the atoms into helical arrangement of medic moments so now I can repeat again the estimates for the 1d band parameters so here as I said the bandwidth is w it's overlapped between the D levels and therefore the bare velocity would be the distance between the atoms times the bandwidth so it will be huge actually but once again we look at states which occupy the energy space below the gap for these states the time electron spins on D level is small it's delta over gamma and that effectively reduces the hopping attempt frequency so effectively it reduces the Fermi velocity down by a factor delta or gamma so now we're getting in business so the effective for most is low and for now as you see I considered hubs between the irons and I neglected the hubs through the ball that I considered before when I looked at cheaper states now the rest is the same the induced superconductivity the gap is over the order of delta times alpha and the ratio which determines the clearance length is again delta independent so it depends on the parameter of the chain on the widths of the band arranged by the metric atoms on the hybridization parameter and spin orbit angle so this estimate actually fits well with the one I had before if the widths of the band for irons is of the order of the widths of the levels of individual iron atoms so that basically explains why the correlation lengths which also determines the shiba state decay is remain short also in the limit of overlapping atoms now just a couple of slides telling that we can go we went beyond the estimates so the idea is that we basically repeat what Anderson and and shiba did but for a chain of atoms so we start with the Hamiltonian that includes the d atoms superconductor and hybridization apply minfield then introduce hybridization to form a matrix equation for the green functions and it's eight by eight matrix because we include the d states then we can exclude the d states and reduce it to four four by four by four n by four a matrix for the itinerant electrons and that will introduce the self-energy into the green functions for the itinerant electrons and the self-energy has a large a strong frequency dispersion and small frequencies it's this gamma over delta term that in terms suppresses the z factor in the green function and that's basically what renormalizes permeability from vf down to vf times delta over gamma so that basically the mechanics how the delta drops out from final answers so and then the rest is kind of very simple technical thing you reduce this system further to two n by two n by by proper projection on polarized states spin polarized states and then discrete discrete Fourier transform yields a dispersion relation which has all the terms that I kind of alluded to before so there is a term corresponding to the hubs along the chain there is a term that corresponds to hubs through a superconductor there's no local hubs which correspond to the k one over r of super states there is renormalization of velocity coming from this term and finally the induced gap coming from this one so all the ingredients that could be understood on the conveying level are in the exact well been filled exact equation and using it one can find spectrum excitations density of states then write it in real space and find marana states so that's basically the end of the kind of scientific part and here is a just picture so one can get plot the diagram in the same basically in the same variables vertical axis is super state position and this is the bandwidths of overlapping irons and again there are two topological and trivial sectors with topology plus and minus one the spectrum of the particles in the one-dimensional band also be a signature of the physics that we discussed so the flat the flat parts of the band actually correspond to unescapability of the shiba level from the from the gap so this approach of the level to the gap comes is is responsible for the flat part of the spectrum there are some peculiar points in the spectrum coming from the shape of the band coming from the long range hops and the gap is the new gap is proportional to delta times spin orbit angle now if you plot this state out of this spectrum one gets away from the edge this blue curve and these additional peaks there are quite a few of them come from the one-hole singularities that are due to the flat parts of the spectrum due to the long range hops and due to the flattening and at the edge there is indeed a delta function peak it's broadened artificially here now again we just use the parameters from dft calculation that was part of the experimental uh slash theory paper um uh from yasdani and mcdonald's groups and if you use these parameters we find indeed that the decay uh on several orders of magnitude comes uh on a distance of order of four interiron distances so this basically the main conclusion that this decays indeed is insensitive uh to the gap physics superconductor and it's just because of the velocity of normalization yep excellent so uh looking ahead actually uh in experiments in the system experiments uh the the as i have shown in the beginning the peaks are pretty broad and uh what we thought as a suggestion to future experiments is to switch to superconducting tips the density of sensitive superconductor is is a sharp feature and one may try to scan with a sharp feature uh with zero bias zero energy states uh and because you you scan a singularity with a singularity there must be much sharper uh conduction uh response uh and uh what we predict is some unusual shape of the conductance peak with some universal value at the top and some width that depends in a non-political way on the tip to uh substrate tunneling so uh you can read if you want it's published already well it's it's already uh so the main conclusions are that uh first that the decay uh of the uh of the marana state actually actually can be quite short uh because of the nature of states uh in the induced superconductor and in future it would be nice to have measurements with uh superconducting STM tip so the the motivation for this work actually was coming from experiment obviously but uh i was primed to this kind of questions uh because i was interested uh in the effects of impurities onto uh topological insulators and uh this interest came in part from discussions with Boris uh about two to a half years ago and it was very useful uh insightful i would say so i just want to finish uh telling that i'm looking forward to uh more insights from Boris in science uh in nature and uh to more uh inspiring conversations uh and and most of all um i'm looking forward to further guidance uh an example uh in the wonderful lifestyle so with that thank you very much happy birthday thanks very much Leonid uh if there are any questions comments uh just bring it up in our classroom as it's being recorded uh yeah i had a question about you use a one-dimensional model but in reality the the atoms are on a 2d surface and can cannot the the wave function can they not localize in the in the direction perpendicular to the chain and and you know on a on a much longer i can understand why along the chain what you say is correct but how about the direction perpendicular to the chain on the surface of the or in the superconductor they also localize in a much longer much shorter direction uh right yeah the good question uh yes so they we don't have we don't have any physical formulas uh but the localization in the in the direction perpendicular is also uh suppressed by the same factor uh so it's not sign in in the in that direction as well so that's uh come from the same basically from the revolution velocity i actually have two questions one is um what about uh andreov uh processes for uh that in principle can give finite widths for shiba states like a pair can go one electron to one uh state and another to another and bring some widths and it doesn't look that it is small i mean shiba or mayrana uh no if you have two impurities yes uh then uh there is andreov process which gives finite widths you can put one and yeah uh finite widths for for shiba state yes absolutely so uh is it taken into account and your yeah good so the essentially if you want the bandwidths uh of shiba states that is formed by a chain is this process uh it is andreov process uh no but uh there is also a real process i mean there is finite lifetime uh well in a chain i mean where would you go i mean uh there is a gap in superconductor right so you can only take a pair from electrons uh localized onto onto shiba states and put into superconductor with energy equals zero right because there is only condensate there so therefore there is no uh lifetime uh there is no bus if you wish uh available uh only if you couple this chain to something else to normal leads but but is there a conservation of total spin or something like that which uh is in your Hamiltonian uh no there is no conservation spin orbit so spin is not there sure but but maybe the same the main thing is basically there is a gap so there is no states available outside this chain that's that's the main thing and second question is about Bogolev of Dazhen uh how uh good is this approximation in the vicinity of the uh impurity yeah it's not good at all yeah we cannot go to to the atomic distance of course yeah so some kfr over r is not valid or atomic distances so the um the experiment experimental trace from yes danya you showed showed quite a bit of both is that temperature strongly temperature dependent or is that zero temperature yeah so thanks for the question it's uh uh there was lots of commotion about it so uh probably yes and the temperature in his experiment uh was uh a buff for Kelvin actually lead allows it because lead has tc 10 something so so they weren't called at all so probably it's temperature now uh there are better not for marines but for shiba states actually now there are better experiments from uh university from free university in berlin kassarina franke this one opens colleague and actually she does use the uh superconducting tip so this is that you can even evade i mean evade temperature even fine temperature you you adjust the resonances so that uh and apparently the natural widths of the levels is is much smaller and it's good and well it's it's pro it's good for science whether it's good for the confirmation it's not clear because there is a forest of states i mean for two impurities there are way more states that you would expect you know from the considerations another question comment oh in that case we was i can amplify your concern so you use a mean field approximation for this innocent type impurities is this just the justification for that that you just know these atoms are magnetic or would there be anything beyond mean field um yeah so uh that's it's also a kind of pressing question so in principle in one over s you can justify it for large spins uh and uh that's basically what we had in our minds without checking much uh now if you're taking into account uh finite value of spin and and even do the projection uh on the lower energy space uh so that you assume that atom does retain its spin but then due to the quantum system you have conda effect uh now my understanding that it's not that terrible because basically uh you can think of replacing exchange constant well j new by conda temperature and the estimates probably remain the same but fully quantum mechanical problem uh for single ship estates there are lots of papers now not lots but okay or out of 10 um and there are some experiments actually uh but for even for for couple of impurities uh so that says that two impurity problem that is well known for normal metal is not uh discussed yet for superconductor so the fully quantum problem is not addressed yet uh even at the level when you have spins and and and putting uh kind of live d levels is even further uh step up okay well thank you very much we now have a break and we resume at 11 30 thank you very much