 Thanks very much. It's it's a real pleasure and honor to be here as part of this special event So many thanks to the organizers for the invitation. So what I'm going to describe is some theoretical predictions for an effect that arises if you combine a common aspect of mesoscopic systems namely Coulomb blockade with the concept of Myranna fermions, which are believed to appear in systems involving topological superconductors. In particular, I want to explain how this combination of Coulomb charging and Myranna fermions can lead to an interesting form of condo effect, which is stabilized by the same topological considerations that stabilize the existence of the Myranna fermions. So we refer to it as this topological condo effect. So this is work that was done by Benjamin Berry when he was a research fellow at Cambridge and it's a pleasure to acknowledge the enormous contributions that he made all essentially all of the key insights and certainly all of the calculations were made by Benjamin who's since moved on to a permanent position at in Birmingham, but we also benefited enormously from some collaboration with the group of David Logan who did comprehensive numerical renormalization group study of the resulting condo effect from the model that I'll describe. I should say that since the original publication that I'll describe here, there's been a lot of follow-up work from Benjamin, Alex Altland and Reinhard Eger and Alexis Fellig, which has really pinned on and expanded on the nature of this the non-firm liquid condo effect that appears in this model. I wouldn't have time to talk about this these developments today. Rather, I'll focus on the key ideas in our original paper and I do that because these ideas and the ingredients that are required in order to get there seem to be something that could now possibly be achieved in experiments owing to the advances in the groups of Leo Calvinhove and our chairman. So the idea is simple enough that one can explain the key ideas in one slide. And so it's based on two facts which are well established. One is the condo effect that if one takes itinerant electrons and couple those to a quantum, a localized quantum degree of freedom, say a spin-a-half impurity that has degenerate levels, then at low temperatures one gets strong interactions developing strong correlations leading to this condo effect and one gets interesting and subtle temperature dependencies of the scattering rate and the conduction arising from the quantum dynamics of the spin as this spin gets flipped by the conduction electrons. A second fact is coming more recently from theory and studies of topological superconductors as I'll expand on in a moment. These are expected to lead to Majorana fermions or Majorana zero modes, which lead to Grønsted degeneracies or degeneracies coming from the topology of these systems and these topological degeneracies can be thought of as some two-level systems or some qubits with degeneracy protected by the topology and so we have another route in which one can get to localize two-level systems which look somewhat like spins and the question that we asked and what I'll describe in this talk is what happens if you know a couple itinerant electrons not to regular spins but to these these non-local spins coming from Majorana fermions that form these topological qubits and as I explained this leads to an interesting condo effect stabilized by the topology strong correlations arise just as they did in the regular condo effect and here again there'll be interesting temperature dependencies of the transport properties and as I'll emphasize these transport signatures really this temperature dependence really shows that the If you see this in the experiment, this would really show that you have some non-local content degree of freedom described by this topological qubit So that's the key idea and I'll try and sketch through flesh out this in a bit more detail So the outline is the following. I'll give a brief introduction to Majorana fermions just to Highlight the ideas that are needed for the talk. I'll quickly say a couple of words about Condo effect and then I'll spend most of the time just explaining how one can map this model of Majoranas with coolant blockade onto a something it looks like a familiar condo model And then I'll describe the predictions for the electrical transport properties so the Majorana fermions was to introduce these it's helpful to note that one For any fermionic system described by some fermionic destruction and creation operators in single particle states labeled by J one can always construct the sums and differences of these of these operators and The operator and their adjoint to make a self-adjoint operators, which are these gammas, which are the Majorana fermions often said to be their own antiparticle simply because this the Gamma is equal to the adjoint gamma dagger So this of course one can always just do as a for mathematical convenience it may be it may be helpful to work in this basis of Majorana operators rather than the original fermions, but the recent interest in Majorana fermions comes about because the actual quasi particle excitations of certain topological Superconductors can force this interpretation of Majorana fermions on you these topological superconductors essentially come down to P-wave superfluids of spin polarized electrons in one shape or form and the the sort of fractionalization that I mean is that you may have some quasi particle Exitation which is a fermionic object, and it would be something which is Delocalized so if you were to work in terms of these C's let's say C was the operator to create an annihilate one of Fermion in this mode then it would be something that's especially delocalized However, if you form the Majorana's the sums and differences then you get one Majorana Especially localized at one end and another Majorana especially localized at the other so the special locality of the Majorana's Is a real physical? Consequence and it forces you to think in terms of Majorana's rather than in terms of the conventional fermionic operators So these This is well established. This is well established in certain idealized models of spin polarized p-wave Superconductors, but an important advance was to realize that one can also get the same physics in more prosaic settings namely if you take Some nanowars semiconducting nanowars with strong spin orbit coupling and Proximetize them with S wave superconductor and then apply a magnetic field so you break time reversal symmetry and have some spin polarized electrons then this behaves as a effectively as a P-wave Superconductor and at the ends of this one expects there to appear these Majorana Exitations and indeed this is what inspired these experiments from the DELF group reported some time ago where they showed evidence from tunnel spectra that they're of Majorana mode at the end of Proximetized superconductor and its junction to a normal wire So I don't want to go into specific Realizations the what I'll describe is very general But let me just say a couple more words about the generic features of these Majorana fermions and what stabilizes them And essentially it comes down to the fact that in superconducting systems described by the boggling of Degen Hamiltonian then the spectrum has a symmetry between plus and minus energy Every mode at plus energy comes with a mode of minus energy and this just reflects the fact that the spectrum the eigenstates of the Spectrum described both the creation and destruction operators for for quasi particles And so the plus E and minus E levels just correspond to the creation of destruction Operators for the quasi particles, but then this leaves an interesting possibility that can appear Which is that you can in principle have a level that appears exactly at zero energy so this Indeed this is what happens if we take a system where we have one of these Topological superconductors and we consider just the half half of it and it's boundary to a vacuum So we solve for a problem where there is a bulk gap superconductor on the left hand side on the right hand side There's some say normal region, which is not not topological at all And in the boundary between these one finds that well if you solve for this system One finds a spectrum of this form where there's a mode exactly at zero energy Corresponded to a minor animode just sitting here So this is a situation which is topologically protected because now you can see that if you have such a level If you were to perturb the system by some local Gates or whatever you like There's no way in which you this single level can move away from zero and maintain this spectral symmetry So this this protection this this symmetry protection of this level at zero energy is really what underpins the Topological protection of the mayranas Now of course in practice, so I cheated a bit by saying we'll just think about one end of the wire Of course, there's another ends and at the other end there'll be another mayrana And if these two mayranas couple to eat so there'll be two such levels in the entire system And I say couple to each other then then indeed there were two levels that can move away from zero energy However, if this is a gap superconductor and the mayranas are far enough apart Then this splitting will be exponentially small in the separation And so this this this mode that one can construct these two mode two mayranas will be at Zero energy to exponential accuracy, but still it can be helpful to think of this pair as forming a regular fermionic mode With the creation and annihilation operator in the usual way and the special feature now is that There'll be exactly zero energy cost to add a fermion in this mode. And again, as I said if we break it into the Gamma one and gamma two then these will correspond to modes which are especially localized at the two ends So this is what underlies this idea of topological qubits this these these two states of unoccupied or occupied occupied fermion in this mode these two degenerate states are the two states of some qubit which is protected due to this robustness through the topology that I described before and Of course, there have been many works which have suggested how one might extend to arrays of these systems and Perform qubit operations on this register of topological modes, but I won't be Considering this. I'll just be interested in these qubits and their description in terms of mayranas so That with that background in mayranas. Let me just give a one-slide Comment on the condo effect So the condo effect as you all know arises when one has an impurity spin coupled to some conduction electrons And so we have a coupling which here for anti-farm magnetic coupling has some exchange coupling j with the spin operator on the impurity Coupling to the spin density of the conduction electrons at the origin which is where the impurity sits And as you know very well at high temperatures if we have weak coupling Then you get weak scattering of the conduction electrons off the impurity spin However, as you load the lower the temperature the effective coupling strength gets stronger and stronger and diverges and Until at in the limit of zero temperature one has a screening of the spin with the formation of a bond state of this well It's been a half impurity with a spin a half conduction electron leaving a non magnetic state and a Fermi liquid with this Spin taken out of the problem now. There are variants which I'll come back to involving over screening which are commonly Discussed in multi-channel variants and this will be important in what I discussed later, but for now I just Put this up to remind you of the form of the coupling that we need for the condo physics So that's all I want to say by way of outline But I want to focus on how we can go from a Situation where we have some myrannas and topological qubits to make something that looks like a condo model So this is first we want to make our spin from these myrannas And so we're going to use this topological degeneracy to get some spin degeneracy and our topological qubit will be our Some non-local spin and the essential idea is shown in this in this diagram So this is meant to represent a grain of superconductor So this whole thing is some small grain of superconductor, which is going to proximitize some nanowires And the grain is small enough that it has a charging energy EC which is going to be a large energy for for our considerations with this So this is where the interactions and correlations come in from just cool and block head on a small grain with a charging energy EC And what we'll need is we want to have four myrannas and there are various ways in which there are various Realizations which could give rise to this but for the sake of argument Let's imagine that that we were thinking of this these this nanowires set up and then we would have two two nanowires Proximetized by the same this one piece of superconductor and then at the ends of these nanowires So the the end between where they maybe this should be closer to the end here the end between where we have a Proximetized nanowire and something which is normal or non topological. There will be a one of these myranna modes And there will be four of them gamma one through to gamma four and it'll be helpful to just to count the states It'll be helpful to think of pairing these up one and two to make a regular from you on C12 And also pairing these two guys to make a regular from you on C3 four now the The point of having these four Myrannas as it gives rise to a grind state degeneracy of two on the island So to see that that source just think about the charging energy So the charging energy will say that at a given bias Background charge q there'll be some favorite number of electrons that sits on this grin that says and zero and for the sake of argument Let's imagine that and zero is even then one can imagine Well, there's one grind state where we have a whole set of Cooper pairs Which are just in the superconducting states and we don't occupy this zero energy these zero energy modes of the Associate with the myrannas however, there's an exactly degenerate state at least within this topologically protected or topological space There's an exactly degenerate state where we take one Cooper pair out of the superconductor Split it into two fermions and we put one fermion into the mode Into the fermionic mode formed from gamma one and gamma two and one fermion into the mode formed from gamma three and gamma four So this is represented here as the occupation numbers the two states are either zero zero where both of these two Fermionic modes were both unoccupied and there's an exactly degenerate state where we split this Cooper pair And we put one fermion into each of these and so that's denoted one one So these two states to degenerate states appear here if we had odd number of electrons Then say we change q and we made n zero plus one the grind state then the same There's still a two-fold degeneracy, but now it would be occupations of zero one and one zero so the setup with four myrannas is one where you're guaranteed that if if you're in this cool and block hid regime the Grind state is two-fold degenerate and that two-fold degeneracy is going to be these these two states are gonna We're gonna think of it as the spin up and spin down of our impurity So now we have to couple those two conduction electrons And there's a very natural way in which that's done Which is that well in this nano are set up Though all these wires can extend further out just as in the experiments that have been done before I'm one can one can induce one can have a tunneling point contacts between a normal wire normal semiconducting wire out here and the myranna mode nearest to it So so here The natural thing to do is to couple these couple wires to these Myrannas now there are four of them in the system and we could have coupled a fourth wire as well Here I'll focus on the case of where we just coupled three wires as you'll see Three wires turns out to be the minimal setting in which one gets something interesting and non-trivial And so that will be enough for the purposes of this talk But of course there is a variant of this where we allow there to be a fourth wire coupled as well Okay, so let's Describe the coupling well the overall Hamiltonian then is the Hamiltonian for the leads which are just free Electrons and I should emphasize that these leads here these leads in Typical setups one should think the leads is really being spin polarized And this these in these setups you break time reversal symmetry or you you have spin polarized fermions in the in the Topological superconductor the the normal wires out here what can think of them as being spin polarized Leads as well, so so we'll have a lead Hamiltonian just describing one component of spin polarized fermions For each of the well three leads that are coupled to the device there was a charging energy I discussed before which gives us parabola and selected out the favorite number of particles And then we have the coupling I said the coupling comes from these tunnelling from from a lead electron So we destroy a fermion at the end of lead J and we tunnel into the Majorana mode which is nearest So that's the gamma J is the with some amplitude TJ for the tunnel amplitude of each of these points and the only other thing we have to Be careful with is to take care that when an electron left to lead and went on to the grin Then we change the number of fermions on the grin so this Exponential of the phase of the superconductor is just the operator that that shifts the number of particles on the grin So this just counts the particles that go on and go off But the details of this don't really matter. It is simple We're just going to look to second order in this coupling and just look at the virtual hopping where so for example We start with n zero particles on the grin We virtually we consider the process where an electron comes off lead J Acts with gamma J increases a number of particles by one So we pay an energy EC and then we jump back down Perhaps with a different on a different lead I acting with gamma I and psi I dagger and so we generate a term of the form shown here bilinear in the Majoranas and bilinear in the lead electrons and with an amplitude which is just T squared over the charging energy So this is our effective coupling and now what I want to do is to to write this in a form that makes it look More familiar more familiar as a condo type coupling And so if you bear with me that is explained on this one slide So this is the term I wrote down before these handling these spin polarized electrons into the into these my anus and the The nice feature and this is the what what makes three leads particularly convenient for Description is that if we have three my anus Gamma one two and three then the bilinear is of these my anus Form a representation of the poly matrices for a spin a half particle So these just give us these bilinear as we can just replace as by the operators Which are the poly matrices acting on our or impurity spin or our two-level system, which is this topological qubit So so that's we'll let us write this gamma i gamma j in terms of a spin operator But to to write to think about let's think about the the lead electrons and to do that I'll focus on the case of just the one two coupling lambda one two which is a symmetric Lambda two one is equal to lambda one two using the fact that the my anus Anti-commute we can write the one two term in the form shown here, and then I use this Representation of the bilinear gamma one gamma two is just i times Is i times sigma three appearing here? and so we have this combination appearing for the lead electrons which of course we can just write as a a scalar product of a three component vector of the lead electrons at the end point with some three-by-three metrics which takes a form shown here and What one quickly realizes when looking at this and looking at the other terms is that the matrix that appears is nothing But a generator of rotations for a spin one object, and so the coupling term here One can write just as lambda one two plus sigma three and j three so this is like Sigma three and j three for the spin-a-half impurity and a spin one object formed from the three lead electrons and you can do this of course for all three couplings and the The nice feature is that one just it just reduces to this coupling of the spin-a-half operator with a spin-one operator so the the system although it started from just orbital degrees of freedom and my eranas it ends up looking like a Condo model where we have a spin-a-half impurity coupled to a single channel of spin one conduction electrons Okay, just just to emphasize the conduction electrons themselves have got no spin the the three components of this j equals one are just the three separate leads and the it's The beautiful feature of the form of the coupling that leads to this feature that one can write the coupling Just as something it looks like a spin coupling of the impurity to some spin or angle momentum in the leads so this is a setup that has been Noticed in the past and has been analyzed and it's an example of an over screen condo problem because well I should emphasize this this is an anti-pheromagnetic coupling as well So the anti it turns out to be anti-pheromagnetic. So if you were to consider a very strong coupling limit You you're in a situation where the spin-a-half object if it if it were to bind a spin one lead electron in the in an anti-pheromagnetic way then you'd still be left with a net spin and so this doesn't quench the spin completely and leave you with a Long degenerate state you're left with a degenerate state that can continue that can still be a screen For the screen and this these over screen condo problems lead to non-fermic liquid behavior, which was explored in These theoretical papers in other contexts But let me just emphasize the spin is there are nice features here the spin is distributed over the different the different Leads so for example by measuring conductance between different leads you can measure sort of anisotropy of the spin-spin coupling in this problem You really have you can just buy you know specially Attach you know looking specially you're actually you can do make measurements that tell you about the internal coupling of these of these spins and the other thing that is clear here is that This makes it clear why three leads was minimal We needed three leads in order to get three poly matrices if there are only two leads If we need to lead site gamma 1 and gamma 2 then we'd only get sigma 3 We wouldn't have sigma 1 and sigma 2 so there'd be no spin flip processes And so if we just had two gammas, then this would just be say sigma 3 and you could diagonalize the system in for fixed Sigma 3 and there would be no spin flips and therefore Whatever whatever effects that come from this sort of condo physics and which I'm about to describe They have to all disappear if you disconnect one of these leads So if you've got three leads attached you'll get some interesting non-fermic liquid condo if you disconnect one Then all of that has to go away. So this is a very nice and clear sign of the non-local nature of the of this The qubit that's responsible for the condo So let me in the last few minutes just outline the predictions that We expect for the transport properties I'll focus on the two terminal conductance the DC conductance where we apply a voltage say v1 to lead 1 and We grind the other two leads and we just measure the currents I to that flows out of lead 2 So we have a conductance g 2 1 I 2 over v 1 and so this conductance of course It's the transfer of an electron from this lead to that lead So even at lowest order it comes if you just work in Fermi's golden rule It will come about from this the square the module squared of this coefficient Which it takes a particle from say lead J to lead I So just the high temperature limit just in perturbation theory You'll just get from his golden rule. It'll go like lambda 1 2 squared But as in a regular condo as you lower the temperature there Well, there are corrections to this which are Logrithmically divergent as you lower the temperature and these can all be resummed Using per man scaling as in regular condo. You get the same form of condo flow and you find that the This coupling grows logarithmically as you lower temperature leading to Growth in the conductance, which is like 1 over log squared just as in regular condo So so here the flow Just to emphasize the flow sort of is towards Isotropy if you start with with different couplings in the different leads Then they they flow towards more isotropic coupling and so that's meant to be represented here by these The different the different conductances between different leads may be very different because of the different tunnel couplings But under this flow they should approach each other as they they all flow to be Isotropic now, of course the question this only applies Down to temperatures of order the condo Temperature below that this perturbation approach doesn't apply and you have to do something non-perturbative And this has been tackled in various ways now So in our in the original work Benjamin did this by applying the Conformal field theory approach to identify the strong coupling fixed point and to show its stability to the perturbations that appear in this Hamiltonian but since then it's also been while he and Outlander-Negger have shown how to analyze this in more convention conventional bosonization approach and as I said there's numerical renormalization group results which Show the same results, but all of these are an agreement I won't go through the details but just tell you the answer and the answer is that the system flows to a strong coupling fixed point which is stable a robust against any anisotropies in the model that has written and this Non-firmly liquid fixed point is characterized by the fact that at low temperatures there are corrections to the conductance which a scale with a Fractional power of temperature and the nice feature here is that we didn't require any fine-tuning The we didn't the system even if you've got different couplings between the different leads the system flows to the isotropic point and is robust to Where there are no there are no relevant perturbations away from it? This I mentioned is just to contrast it with a more conventional Or another more familiar form of Overscreen condor, which is the multi-channel Condor problem where you may have for example two Different channels with a spin a half electrons coupling to a spin a half impurity and here if You can choose things so the coupling between one channel Well, but the coupling between each channel and the impurity is Identical so the two couplings of the same then the system shows non-firmly liquid behavior with the square root of T Corrections to the conductance at low temperature, but if the if there's anisotropy in the coupling between Say channel one is more strongly coupled than channel two then the system will flow away from this to Conventional Fermi liquid fixed point of one type or another Here we didn't require any fine-tuning the model. I wrote down if we play around with if you any choice of the parameters of your is it stable this strong this non-firmly liquid fixed point is stable So the so that's why one feature another feature is the load the saturating value of the conductance is some Universal value two-thirds e squared over itch So at low temperatures, it's isotropic So again is a nice feature that if we have different conductances at high temperature the conductances should flow to to being isotropic at low temperatures One just one comment on the numerical value this two-thirds is interesting in the sense It was remember that this is defined as the current flowing out of lead to If we there's also the same current flowing out of lead three so the total current flowing down lead one is actually twice this It's four-thirds e squared over itch so the conductance of this wire is four-thirds e squared over itch And remember the wire is spin polarized fermions. So you may think it's it's curious that there could be You could have a conductance larger than e squared over itch in a spin polarized wire Of course one way you can do that it was with superconductivity around is by having under F reflection where a particle comes in Cooper pair is absorbed and a hole is reflected. However, just be note here that Because we're working in a mesoscopic system in the blockaded regime There's no possibility to absorb a cooper pair But what month so what must be happening is that whenever the cooper pair comes in it's immediately ejected as a pair of electrons and so this the scale of this of this Conductance shows that there must be correlated Andrea reflection processes where an electron comes in a hole is sent out and two Electrons are sent out in two other leads. So there are interesting characteristics of the noise arising from this Correlated Andrea reflection so So I all I've commented on are the limiting cases of course is very important from comparison with experiment to look at to look at the the crossover between these asymptotic limits and so for that David Logan's group did a very comprehensive numerical renormalization group study The results are shown here Different temperatures the conductance is a function of frequency. Well here showing the low frequency and high frequency asymptotes and confirming Essentially it confirms the asymptotic behavior They confirm the robustness against the anisotropy of the tunneling amplitudes and deduce the crossover curve Here also shown here the DC crossover curve is function of temperature one thing just to to note from this and I think this is common to also to other condo problems is that the It's really rather hard to To to get into the limits the asymptotic limits of high temperature or low temperature You have to be a couple of orders of magnitude above the condo temperature or I'm one or two orders of magnitude below the condo temperature Which of course is very difficult when in a narrow temperature range accessible and experiments so So actually in the experiment to fit to experiment one probably does need to have the the full crossover curve in order to see To make quantitative comparisons to to the theory So this so I've told you that the system is completely robust and you have a this stable topological condo Effect, but of course there must be something that can kill it And so I want to just turn to comment on what does kill it and the thing that kills it is the same thing that kills the The protection of the Majorana modes namely if you have if the Majorana modes can couple if they're if a tunnel couple between each other then this Leads to a splitting just as that leads to a splitting of the of the degeneracy of the Of the topological qubit that will that will destroy this this condo effect And that of course makes perfect sense because if we have enough tunnel splitting then we shouldn't think in terms of Majorana's we just have electrons. There's nothing special. We'll just get we should just get conventional Coupling to conventional fermions, and there's no there's no special degree of freedom to to give us any condo So this this arises So what was special in our model was we had tunnel coupling just from lead one to gamma one lead two to gamma two And lead three to gamma three if you allow for something like a tunnel coupling from gamma one to gamma two Or if lead one can tunnel into gamma two then this this does Introduce a relevant perturbation it breaks an effective time reversal invariance in the problem And you can see that just from the fact that the tunnel coupling gamma one gamma two I can write as Sigma three in this polymetrics Representation so it looks just like a Zamen field and of course if you apply a Zamen field you'll you'll destroy that the You'll you'll gap out the condo effect, and you'll destroy Prevent the spin flips required for it, and so this introduced a new crossover scale and indeed if you and the numerical renormalization group studies they they did a very careful study to show to look at these Crossovers, and this has also been worked out not for the conductance for it, but for the thermodynamic properties and very nice work by Outland Barry Egger and Svelik where they've also been able to show this using the Better and that's results Okay, so that's let me just summarize so the Message I want to get across is if you have itinerant electrons and my run a firm couple to my run if fermions Then if you introduce cool and blockade to have an interacting system Then the system shows this interesting topological Condo effect now. It's it's one nice theoretical aspect is that you get this robust realization of non-fermic liquid Condo physics, but from the practical point of view of where we are with experiments I think it may be Most interesting just to notice that it gives a very clear way in which one can see the fact that you have a quantum non-local qubit or a non-local degree of freedom In these topological systems and the the clear evidence of non locality comes from the fact that well whatever the Functional form of this upturn in the conductance is a function of temperature So let's say we measure the conductance between lead lead one and lead two and we get some interesting temperature dependence What I've said is if you turn off if we cut lead three which apparently is not doing anything in this transport If you just if you just disconnect it then all of this enhancement disappears and you just go down to something Which is just standard temperature independent? tunneling through the through the system so this this And loss of the enhancement when you disconnect lead three. I think it's a very clear Demonstration of this a non-local quantum degree of freedom in a simple transport experiment So with that I'll stop and thank you for your attention