 Thank you very much to the organizers first for inviting me here, it's a real pleasure to be here. And so I want to talk about topological properties of multi-terminal chosen junctions. Today I start by thanking my co-workers. So all of this work has then been done together with my colleague in Grenoble Manuel, we started a couple of years ago with Yuli Nassarro, who was visiting at the time and a postdoc Roman Rivar. And the most recent work in Brokers that I'm going to talk about today is then in collaboration with Pianata, who was a postdoc and is now a permanent researcher in Grenoble. So the idea is that I'm not, I don't want to talk about real materials, but what I'd call synthetic materials. And then this we want to use multi-terminal chosen junctions to make an artificial material. And I'll explain you the analogy in the first slide. The advantage may be being that it's more, it's simpler to control than a real material and to engineer the properties that you want to look for. So the idea is basically if you create just a simple chosen junction, meaning you have two superconductors that are separated by some non-superconducting material, the superconductors have a gap. But in the junction, you may form bound states with an energy that is smaller than the gap and the energy of this bound states actually depends on the phase difference between your two superconductors. So you have something like a band structure where you have an energy band that is periodic in this superconducting phase difference. So if you have just the two-terminal chosen junction, it's a one-dimensional band structure. You have one phase difference. If somehow you manage to connect multiple superconductors to some same central non-superconducting region, the bound states in this central area may depend on all of the phase difference. So if you have n terminals, you have n minus one phase difference. So you have the n minus one-dimensional band structure in this material. So that's the analogy we want to use. The entree of spectrum in a chosen junction is our pseudo band structure, where the phase differences between the superconductors play the role of quasi-momentum. So again, n terminals makes an n minus one-dimensional band structure. Of course, these are not the same as quasi-momentum because they're external parameters. So in some sense, we do single particle physics, we don't have a firmacy. We have a given value of phases at a given point. And if you want to probe something for the entire band structure, we have to vary the applied phase differences to the material. So what can we do with that? Well, we can look just as the energy bands, but you can also ask about what we do with neutral materials. Is there some topological properties in these artificial materials? So what we studied in the beginning, and it's the simplest example, actually, is if you look at the four terminal chosen junction, just use conventional superconductors or nothing topological about the materials we use, and look at the Andrea spectrum in such a junction. So you have three phase differences. And what we showed, what can get is an analog of what one might call a by-semi-metal, where you have a topologically protected crossing between two Andrea states. And particularly interesting are the crossings that are at the Fermi level. And the Fermi level is special, again, because we have particle-hole symmetry in this system. So unusual by-semi-metal crossings can happen at any energy. Here particle-hole symmetry makes the Fermi level a special point. So you can have an analog of a by-semi-metal. Now here, even though this looks like a three-dimensional material, you can choose how many phases you actually consider as parameters and you can go down from a three material to a two-dimensional material by just fixing one of these phases and considering that's the mass term. And then if you fix it on either side of such a by-point, is characterized by turn number and this turn number changes as you vary your control phase across this by-pressing. So you can have a two-dimensional turn insulator where the ground state on one side of this crossing will have a finite turn number. And then a finite turn number usually in a 2D system is associated with this edge state and a quantum Hall effect and you can ask where to get the analog here. Note however, we don't have edge states. In some sense, this is an infinite system because the system is really periodic in phi and we don't have to conjugate variable to play with unless we want to play with charging, which we don't do at this point. We showed that even though we don't have edges, we actually can measure the turn number similar as in the quantum Hall effect by measuring a transconductance. In a way that's done somewhat differently, you apply voltages to sweep your entire prenuason, so to vary the phases, so to explore really what happens for all the values of the phases and it turns out that we get a Hall conductance that's proportional to the turn number of this band structure and I'll show this later. So the new work that I want to talk about today as well as while this is an example for topological band structure in Josephson junctions is there more. We know that in real materials, there's a whole host of possible topological phases in different dimensions characterized by different logical invariance. So even there's a table which tells you depending on the symmetries of the system under time reversal of particle hole and a chiral symmetry, what are possible phases in which dimension. So the question is the analog for Josephson junctions and it turns out it's not the same table because even though my say my phase difference are an analog of quasi-inventor, they actually don't behave the same way under these symmetries. So the classification will be different and so we can possibly have different topological phases in different symmetry classes than in real materials. And the other advantage of this realization is that we are not limited to three dimension in principle, we can attach as many terminals as we want to the system and ask for topological phases in four, five, six dimensions. Now something I won't talk about today, but I'll flesh it because if you have a flat band in this cases, if you don't use conventional superconductor, but you make Josephson junction with topological superconductors, the one the Gitaif chains basically that Tikhan talked about in this tutorial, you get a different kind of band pressing which now involves three bands. You have two finite energy bands and the flat band which is basically a single Majorana that's staying at the Fermi level in the three terminal junctions. And what you showed that in this case our 2D band structure is always non-trivial. So in this kind of topological three terminal junctions one would always get a finite transconductor. Now what's the experimental situation? So the things I'm talking about haven't been measured yet, but multi-terminal junctions is something that people can realize and there's more and more groups that are realizing them today. So I'm showing several examples with different materials. The first one is a metallic junction which is probably not the best way to look at the physics we are looking for because the number of Andrea states and therefore the level spacing in this junction depends on the number of channels that conduct the different superconductors. And if we want to probe properties of individual Andrea states we need a level spacing that's sufficiently large. So we want to have a junction with few channels which can hopefully achieve either in two-dimensional materials where you can have fewer channels or even taking wires that connect your superconductors. So here's my outline. I want to say a few words. What is this unfair spectrum and how do we study it? Then briefly review what we saw in conventional four-terminal junctions where there's clear evidence for these topological properties and then come to this classification and possible phases in these systems. So the basic process that leads to the formation of these unfair bound states is Andrea reflection. We have our superconductors which have a gap in the excitation spectrum so there's no quasi particle excitations around the Fermi level. So if you want to transfer a particle into the superconductor these energies that's not possible as a single particle excitation. However you can transfer a cooper pair into the superconductor which means you need to grab two electrons or equivalently you can say you take an electron and leave a hole behind. So you reflect an electron into a hole. This can happen with a single superconductor. It gets more interesting if you have two of them. So now I enclose some normal region between two superconductors. So I have an electron that gets Andrea reflected from this superconductor as a whole. Once the hole hits the other superconductor it gets again Andrea reflected this time extracting a cooper pair and comes back as an electron. So for some of that quantization tells you so now we have a closed loop. If the acquired phase along this closed loop is a multiple of two pi we will actually get a state. And the phase here has different contributions. If the normal region is long enough you have a propagation phase I want to concentrate on a case where the normal region is fairly short and this propagation phase doesn't play a role. So the bin phase comes from the Andrea reflection. There's a term that is just the phase of the superconductor that is picked up and there's an energy dependent term which basically depends on how the eigenstates in the superconductor are mixed of electrons and holes which depends on the efficiency of this Andrea reflection. So without the propagation phase in the normal region you get a very simple equation to solve and you find that you have a bound state that crosses all the way from the gap edge above the thermal level to the gap edge below the thermal level. So you have one Andrea state in this ballistic case which crosses the entire energy gap. Well I say you have one Andrea state that's actually not true. Here I showed electrons moving in one direction. You always if you have just one system you will have the electrons also moving in the other direction. You get a second state from this additional process. So you actually have two states or one doubly degenerate state which can also be very standard this way because the system has particle hole symmetries so an electron state with a positive energy can always be represented as a whole state with negative energies. So either you say you have a one doubly degenerate level or you have these two levels here related by particle hole symmetry that will cross at the thermal level. So here you have a grossing but actually this grossing is not topological at all. As soon as you introduce any back scattering these two states will get coupled and hybridized and a gap will open up. So in the conventional two terminal Joseph's injunctions you won't have any states at the Fermi level. You will always have a gap in the energy spectrum. So what about multi-terminal injunctions? So the way we study them is use a scattering formalism where we actually can separate what happens at the interface with the superconductor where you have Andrea's reflection and some normal region where you just have normal scattering. So that means say you have an electron that comes from one of these superconductors. Once it hits the normal region it can be scattered into any of the other superconducting leads. It depends on the matrix that we have to choose appropriately. Once it comes to the other superconductor it will be Andrea reflected. Then the hole again will be scattered by this normal region and go back to the superconductor. So that means if you look at the wave functions in these links connecting the scattering region to the superconductor you have four components. You have electrons going one way or the other way and you have holes going one way or the other way. The electrons going different directions are connected by normal scattering. Electrons and holes are connected by the Andrea's scattering. So then you can show that you get well if you have all these scattering processes you get go back to the same states actually and again if it's really the same state or like we said before the face of the module of 2 pi you will have a state. So yes if this scattering matrix has an eigenvalue 1 you will form a state in the junction and by using the properties of these two scattering matrix namely that normal scattering only connects electrons with electrons and Andrea's scattering always connects electrons this whole you can this vector which is at least four-dimensional depending on how many channels you have in your junction to something that has half of dimension which is this equation. So here you see again you have this energy dependent phase that I showed you before for the two terminal junctions and now you have all the different phases for the superconductors so dependent from the superconductor it gets reflected except the phase of that superconductor. So we have one equation that will allow us so it depends on phase and energy so we have to find the relation between phase and energy such that the eigenvalue is 1 and that will give us our Andrea's spectrum. And before I said I want to concentrate on short junctions the simplification for short junctions is that this scattering matrix doesn't depend on energy yes. If the gaps would be different in different superconductors then I would have to adjust these phase factors again also for the superconductors so that for each of the superconductors you get the corresponding phase vector is the gap of that superconductor. So the specificity of the spectrum obviously depends on the scattering matrix you choose the generic properties is again it depends on can depend on all of the phase difference it has particle hole symmetry and if we don't include some some spin scattering or spin orbit coupling all these levels will be definitely degenerate. So again if we have four terminals we can have a 3d pseudo van structures if we have two terminals or if we fix one phase difference we have a 2d pseudo van structure. So now what can happen in these band structures well I told you already I might want to look for for vial semimetals so how do I get that is just I have to ask about do I get two bands to cross and actually that's the only thing that or the most simple thing that can happen at a Fermi level again because we have particle hole symmetry I can't just have a single band crossing the Fermi level it necessarily is a crossing if I have a state at the Fermi level. Now if I just look at crossing of two bands I can write an effective low energy Hamiltonian and since it's two band I'll have a two by two Hamiltonian and any two by two Hamiltonian I can express in terms of poly matrices plus some constant term which actually here will be absent because of particle hole symmetry so you can't have an easy one. So that means I have three poly matrices so I have three coefficients I need to tune to zero to get a crossing between these bands. So if I have three phases I have enough parallel meters to tune my three parameters to get such a crossing so that's why in three dimensions one can generally get a crossing without fine tuning of the junction you don't need an additional parameter to get to this crossing and once you have a crossing while there are no additional terms you can add to in your Hamiltonian so you can't split it you can shift it by putting perturbations on these poly matrices but you can't split it and that's because this is a topologically protected crossing actually it has a topological charge which depends on how these parameters depend on your how the coefficients depend on your control parameters can be plus or minus one if it's a linear crossing. Now if you have such a crossing as I said then it tells you also what happens if you take cuts to your three-dimensional band structures so the topological charge is of these white points is related to berry curvature they're actually monopoles of a berry curvature and berry curvature is also what determines the chair numbers of a 2D system so this monopole actually has a consequence that if you take cuts to your 2D to your 3D structure on either side of this point the difference between the chair numbers on either side will be just the charge of this monopole that's enclosed between the two surfaces so that means if you have a white crossing necessarily on one side or the other of this crossing your states will carry a finite chair number on one side it can be trivial but on the other side it will be necessarily non-trivial so that means such a crossing will give topological phase transition if you want and then what you expect is edge states and a quantum hall effect if this was a real material. Now first questions do we can you get that well we can find scattering matrices where you can get sensory crossing if we assume the system to be time reversal invariant there will be at least four crossings if we have crossings because we always have two with the same charge that related by time reversal symmetry and the total topological charge in the entire band structure needs to be zero so that would be at least four points and if you look at the chair numbers by taking cuts the chair number jumps at every time you cross such a white point and you don't know only know that it jumps you also have a reference points actually if your face the control phase is zero the entire system has time reversal symmetry so you know if you have time reversal symmetry the chair number is necessary zero so you know you start here from zero and then you can just consider the charges and really determine the chair number in other parts of the spectrum so that's also why in the end here I'm only only talking about the Andrea state of course my system has more states there's a continuum outside of the gap however as these states don't first the Fermi level their topological content won't change as I change the control parameters so I have this reference point where I know that the system is trivial I can by only looking at this low energy states determine whether the entire system is trivial or non-trivial so well I said you can find scattering matrices actually if you take just random scattering matrices processing time reversal about five of them will have such white points if I take terminals which have more than one channel the number the probability of getting white point goes up very quickly and almost always find white points in these structures but again as I told you before the more levels I have in the junction probably the more difficult it will be to see the properties of these levels so even though the chance of getting the white points is lower when you have fewer channels it might be a better system to observe them in the end so how to observe them well you can do a spectroscopy it has been done in these sections I showed you the entry of levels and just see the crossing but it turns out you can measure the chair number similarly as in the quantum Hall effect namely if you look at the current between different terminals the current is given as the derivative of the Hamiltonian with respect to the phase of the superconductor now if the phases are constant this gives you just your Josephson current which is a paste dependent current if you apply easy voltages your terminals you have the second Josephson relation which tells you that actually the phases won't be constant anymore they will wind the velocity that's proportional to the applied voltage so once you apply voltage just you will explore your entire band structure so if this voltage is sufficiently small you can compute the current by using the instantaneous eigen basis of the Hamiltonian so you just take the states at a given well at the time given by the value of the phase at the time and you can compute the current with this so you get the first term which is just the ac Josephson effect that means you just take your dc current phase relation and use this absolute time dependent phase which gives you an ac current and you get a first correction which is proportional to the velocity of the evolution of the phases and the term that gives the proponent functionality here is the berry curvature so the instantaneous current my Josephson function measures the local berry curvature of my band structure and well if I want to see a turn number I need to average over the entire premium so I need to explore not the local berry curvature but the berry curvature averaged over the entire premium so I can either apply two increments advantages which means I will sweep all the possible phases or I can just apply one voltage and then step the other phase with the magnetic flux so an average the currents over that so I will explore the berry curvature throughout the entire premium zone and that's what gives me in the end this conductance that's proportional to the turn number so there is a way to measure these topological properties of my junctions so before I show you this picture where you have my white points in the band structure and I said this is related to the turn number it is directly related to the strength conductance that you measure in the system of course this is a small signal on top of a large ac current and the idea is really you need to average out this ac signal to get in the end only the dc contribution of this current if I have time in the end well I'll talk about why I'm having not too many chances important and it's actually the same reason why it's always important when you want to want probe topological properties is you're interested in a property of the ground state so if you manipulate your system you need to probe the properties of the ground state and not excite the system and see what happens in the excited dense where you lose this topological projection it's always protected by some gap to an excited state now this was an example and what we've been trying since is to figure out are there other examples well we said in principle with these manual terminal junctions you have infinite possibilities because you can realize arbitrary dimensions this is an analog of something that exists already so what else can we hope to do so the most recent work that's still work in progress is to looking at a complete classification of what are the possible phases I can get in this multi terminal junctions and then see if we can find other examples so this is the table I already showed which classifies the usual topological phases in materials and there are three columns in the beginning so it depends on how this is if the system has time reversal symmetry or not it has particle hole symmetry or not and finally you can get a chiral symmetry which if you have these two symmetries it will just be can just be the combination of the two but even if these two symmetries are absent you can still have a symmetry just under the combination so that in total gives you 10 possibilities because each of these symmetries can in addition to being present or absent square to one or to minus one because you have this tenfold symbol rather than classification and it tells you whether you can or not have a couple actually face with this symmetry and in a given dimension so the examples that Tigran talked about in his tutorial are the ones I'm showing here so in class a where you don't have any symmetries and two dimensions you get a quantum Hall effect which is characterized by a chiral number if you have time reversal symmetry you can get the quantum spin Hall insulator which is characterized by a C2 invariant and the Kittai chain which is a topological superconductor in one dimension again is characterized by a C2 invariant now this table here while is shown for three dimension because it's the dimension of your field materials it can actually be extended to arbitrary dimensions and arbitrary actually it's sufficient to go from zero to seven because after that one has a period of this city there's a what period is t of eight so from that we can conclude the possible phases in all dimensions another thing that can be included as well is a classification of defect if your Hamiltonian doesn't depend only on k but you have a slow dependence on some variable arc characterizing a defect in your system the position so the position position doesn't change sign under any of these symmetries it turns out that this classification holds the same way and the relevant parameter is actually the difference between the total number of spatial dimensions and the number the dimensions of the surface that surrounds your defect so the full table considering chest these symmetries is the one given here now coming to our system and that's the only thing that didn't work when I transferred my presentation to the other computer this should be arrows as well so the mysteries it's the same arrows as here and this one didn't work so while time reversal actually changes the sign of the phases as in a if that would for quasi momentum k the system as a superconductor has particle hole symmetries at a given value of the phases so the phase doesn't change sign on the particle hole symmetry so in that sense it behaves differently from a momentum variable so that means if I want to and as a consequence since the chiral symmetry will be the combination of the two it will also different behave differently under this chiral symmetry so this kind of parameters actually has been studied before for some example by shang and kain we called it an anomalous parameter so it doesn't behave as it should under the combination of these symmetries and there's another study which makes actually a complete classification of systems that have additional symmetries spatial to spatial twofold symmetries where these parameters may or may not behave in this way this is what we can use to extract a classification for our tensor system let me go through this so here's again the full table now there's some cases which are simple so if I only have time reversal I know what are symmetries I don't care about how the system behaves under particle hole symmetry in that case the table is the same because under time reversal symmetry I do get the sign change as I would in the normal case so the rows where I don't have any additional symmetry I can just extract from the usual table there's another case so that would give me these three rows another case that's also sort of simple is when I only have particle hole symmetry if I have particle hole symmetry phi doesn't depend like a guasi momentum but rather like a spatial coordinate but as I said we can include the spatial coordinates in this classification and it's just delta k dk minus dr so the number of spatial dimensions that means in this case I just have to read the table backwards with a minus sign for all the entries so this adds two more lines to my classification so the line d a I can read either way class eight because I don't have the symmetry so I can use this classification or the other and since it's has a period dcp2 it doesn't matter in which direction I read it so I get the same cnd which only have particle hole symmetry so I just read this backwards and I get the two entries so the ones that are more complicated are the glasses that do have a chiral symmetry because those do not enter in any of these cases of the of the usual table so the work by Ishiyoseki and Sato that I mentioned considered systems which do or do not have the usual symmetries but initial in addition have or the two spatial symmetries such as a mirror or twofold rotation and these symmetries not all components of k and r have to depend on the same way so you may have some that behave in the usual ways that it means they don't change sign for k under a unitary symmetry like chiral symmetry or they do change sign under an unitary symmetry such as the particle hole symmetry and then they're the red ones that behave anomalously under these additional symmetries so if we use this classification we can actually apply it for our junctions in the sense that we only have these anomalous parameters so we say all our files behave like this k parallel and all the other parameters just don't exist in our system so that means if I take a look at these classes well the only usual symmetry is time reversal so if I have only time reversal and not particle hole or chiral symmetry the corresponding class would be this one so a3 doesn't have time reversal so it's a bd1 has a time reversal expressed to 1 which corresponds to class a1 if it has a time reversal expressed to minus 1 it corresponds to a2 and then I have this additional symmetry which is determined by these rows so for class a3 I only have one additional unitary symmetry that gives the chiral symmetry whereas for the others four classes I have an additional unitary symmetry which then gives me also the chiral one which is a combination of the two under which this file is an anomalous parameter and then I can go back to their paper and figure out what are the corresponding entries for all of these phases to see what we can get in our system and it actually turns out that three of these additional classes are always trivial so you cannot realize any topological phase within these symmetries whereas the other two have new topological indices so I can add those to the ones I already established so this is the full table of possible topological phases in these multiple element junctions where I have an hematonia that only depends on this variable pipe so the ones I already discussed fall nicely into this table so the churn insulator in conventional junctions is characterized by an index c so a churn number and it's a system that has only particle hold symmetry time reversal needs to be broken um yes no no c is the c is the usual case without spin orbit coupling so c is without spin orbit coupling so it's just a conventional bcs s wave superconductor without spin orbit coupling in that case you get c if you add spin orbit coupling you go to class b and actually you get a 2c classification and we studied our junctions if we add spin orbit coupling perturbatively actually it doesn't change the classification however it goes to 2c I already showed you we have a quantized conductance which is a quantified units of four e squared over h and one of the factors two actually comes just by adding up the two spins and once you have spin orbit coupling you can't consider the two spins separately anymore so you directly get the 2c classification which can contains the two spins bcs so both classes can be described by by what I showed you before okay so the thing we well there's lots of entries to be explored still well actually um well there's nothing in two terminal junctions no interesting gap phases in um two terminal uh in three terminal junctions so with the d phi equal two there's the two cases we already explored I don't know any superconducting examples for a two and c two so I don't know what to do about those there's another entry in a system without symmetry and there's actually um similar proposal or related proposals how to explore these by not using these uh the unfair spectrum of um choices and junctions but networks of choices and junctions where you can play with charges and phases and where you can break all the symmetries and vls these kind of offense structures that would be um characterized by c and x so in some ways this has already been done so the new interesting things that can possibly be explored is from the equal three and and upward and the one we did look at so far is an example of a c two phase now in class um c again I don't know of any superconducting examples for the classes if someone knows them uh let me know I'd be interested in discussing so what's this um c two phase well again um we can just look at four terminal junctions and look at band structures but you can't see the index from the band structure so it's hard to figure out if it's topological or not the easiest way to get at a topological phase is float for a phase transition so if I want to uh have a topological phase in um three dimension I basically have to look at a phase transition in four dimensions or in this one additional control parameter so what I can look for are protected gap closings in five terminal junctions so I have four parameters and then at the end I say one of them is my control parameter and I get a gap topological phase in three dimensions so similar as before 2d churn insulator 3d by the point so in that case we again can look at an effective low um energy Hamiltonian it turns out two by two is not sufficient as we said if we have two by two we have three poly matrices and not more so if we want to get um four um matrices that anti-commute then if it's our spectrum we have to have at least a 40 representation uh for dimension for the structure of this low energy Hamiltonian so that means we need two bands it's not just a single band at first but we need two bands to cross at the framing level so in that case if we have these four matrices that anti-commute our spectrum is just the sum of the squares of the coefficient and we have a linear opening in all of the four directions so one example and I show you later how we obtained it is shown here where you have this crossing at zero you see the two bands the full band structure in the end is more complicated than the bands don't stay um degenerate now if you look at the symmetries in this class actually these four matrices are not the only ones that are allowed where I said for my two by two systems I just have three poly matrices and that's all here if I looked at the system which has particle hole symmetry there's actually 10 for um this four for the um four by four matrices that would respect um particle hole symmetry so in what sense is my crossing protected so let's see what happens if we add these other matrices I won't add all of them we can do the calculation let's just look at an example so the specificity of these new matrices that I didn't add this is that they actually don't anti-commute with all of the matrices that I already have I can't add a fifth matrix that anti-commutes with everyone so the one I add will commute with some of the matrices and because of that actually we don't get new terms this additional mass term or this additional perturbation to my Hamiltonian will um split my foreband crossing in a point into a line of points where two bands cross at the terminal but it won't open up a gap so what's what's shown here sorry in one direction it's here that now I have a line of gap closings between who between two of the bands but if I go sufficiently far away from this line of gap crossings I still have my phase transition between two different topologically different um gap phases um I think my time is almost up so we have two examples how we identified systems that actually have such gap crossings the one is basically what Pierre and I have did is a brute force search between the scattering matrices to find one that locally gets this low energy Hamiltonian and he found some class of matrices that do that I have no idea how and that's the spectrum I showed you we can also make an effective model with quantum dots particularly if we have two spin two dots or two spin degenerate levels the most general low energy Hamiltonian when there's a superconducting gap is considered largest energy scale in the problem is given here and then I can look at how I can tune these different parameters so I have the on-site energies I have tunneling between my two levels they can either be a direct process but that doesn't depend on the phase of the superconductor so if I want to as a tunable parameter I have to take into account elastic cotoning through superconductors and actually I get a fairly complicated effective terminal that allows me to get a phase control of this process and I have Andrea's reflections which as the clear mask which can either happen on the same level or like a process between the two levels and again if I want to control these two independently I can't just use one superconductors but I need several superconductors which are tested phases to control them individually so the quantum dot model gets fairly complicated I need I think I need over 10 superconductors in the end to make this so this is not a realistic model it's just to show that we can devise something that works and in that case we also do get these protected crossings and linear spectrum around them so let me conclude what I showed you today is this analogy between Andrea's spectra and band structures of materials we showed that we can get analogues of some known topological phases and we performed a classification of all possible phases we found one additional example but there are lots of open questions in particular is there a system that would realize this rather than this very unrealistic models that I showed you in the end and also what is the signature of such a C2 invariant rather than just looking at band crossings in these systems so thank you for your attention. Questions please. Internet. Internet I owe you a number. So did you you mentioned at some point spatial symmetries right and I guess with you we would need three terminals to two dimensions we have two phases and yes because it's basically do you have anything like a reality condition can you implement the reality condition like a C2 Z times time reverse condition where your Hamiltonian is real in some basis. I'm not sure you mean that a local time reversal symmetry rather than the usual one. Well it would be a combined time reversal with in momentum systems it would it would leave the momentum invariant okay imposed because this would have this would get your vials in two dimensions. Indeed we didn't look at any symmetries between terminals so spatial symmetries would involve specific properties on the scattering matrices which we didn't look at at all we assume we have only this particle hole and time reversal symmetries and we don't impose special conditions onto the terminals which would probably not make it for best tool in derivations but in the group structure you don't have spatial symmetry stays the same computation and anti computation but with these other symmetries it would be different. Like you get projection I assume if you have additional symmetries you can get different yes but yeah we should discuss that we can look at them. Okay thanks. Have you thought about how the classification has changed by interactions? Uh no I mean in some way as I said we're doing single particle physics we don't have a firm you see what can change or interactions that can be incorporated in different ways. You can think about just having some capacitances which make the which mean that your phases will fluctuate and not be just an externally imposed parameter anymore which would be interesting because then you have a conjugate variable to play with you can think about defects and boundaries. How the spectrum will change with interactions we didn't think about but yeah in the end it's a question of spectrum but I know that there are interesting things happen to the ender spectra and in persons of interactions but no we didn't look at it specifically. Somehow I suspect it's an even simpler problem than this because it's essentially zero dimensional. I mean if you just take a dot with a local with a local repulsion if your central region is the dot you can get in some regions you know how to compute the ender spectra and then you can look at just the band structure again so that's something that can be explored. How do you include disorder into it? Disorder is actually the fact that we have a scattering matrix in the central region so the interesting thing is we don't need any fine tuning of the scattering matrix if we have one realization of the disorder in the system. Oh what I mean how does it enter into the Hamiltonian description. You have basically a Hamiltonian which depends on several conserved quantities. How it will look like when you include the disorder. So we include well we have a system that can be described by a scattering matrix and I get the given scattering matrix will depend on the realization of disorder and from that scattering matrix we construct the Hamiltonian so for some realizations of disorder as you say we would get these white points for others we wouldn't so for different scattering matrices but if we have them if you make small changes to the system there will be robust again because you need to merge two of them but so the parameters of my Hamiltonian will depend on the specific realization of you say the disorder in in this junctions of the specific scattering matrix of the junction. For instance in most standard cases inclusion of the disorder would be like to add some v term or to add some random gauge potential and here there is no obvious way to to modify the Hamiltonian no. I mean we have really the form of the Hamiltonian will depend on what the scattering matrices is and sometimes we do get these states and sometimes we do need this order if the junction is completely ballistic you will never couple all the different terminals anyway so the system needs to be the chaotic or random so you actually connect all the terminals but then the specifics okay so there's no I don't know of a simple way to go directly from the scattering matrix to the low energy Hamiltonian. I think that maybe this question about disorder so basically one can think of Hamiltonian you think about some randomness in eigenvalues of the Hamiltonian but in that description if you have unitary matrix and you can look at eigenvalues of the unitary matrix unitary matrix is disordered in the proper example and there are some relations between basically between the eigenvalues of unitary matrix in the Hamiltonian so this respect you can classify it the same way the unitary matrix one very last question and I want to say that at the end of the session we have yeah I had one question that when you said by putting this random scattering matrices you can get five percent of this while points so I had this question that can you get more than one while points and depending on the chirality of the bands you can get probably channel number of more than one if you have a well if I have a as I said the the if my system has time reversal invariance and and the the random scattering matrices I show where for time reversal invariant systems if I get a while point I get at least four while points to get something which has a higher churn number in the end I cut it as a higher churn number than one I need more than one channel so if I have cutting matrices in some higher dimensional space I get easily by points and I can get churn numbers that can go up to whatever I like but I have a dense and we have spectrum so it will be much harder to see in the end that I actually did realize them but yes okay let's end Julia again question online how do you observe the edge states we don't have edges in the system actually it's a zero-dimensional system so there's the five variable is strictly parroted if we don't include interaction effects if we play with charges which is our conjugate two phase and superconductors we would have an inhomachines system however I don't know how to engineer a sharp edge and that's what's needed to get a real edge state so we don't have edge states so we basically it in some ways it's more similar to topological pamp so we just broke the interior internal structure and don't play with the edges to observe the topological properties thank you right the second