 It's a pleasure to be here and I should tell you that Boris helped us very much when we started our institute in Dresden because at the very beginning, you know, everything was very incomplete and not very inviting and provisionally and in a phase like that you need support of good friends that start to get the thing moving and Boris helped very much by organizing a very excellent workshop and when he did that I think many people thought well place cannot be too bad if Boris is going there so that helped us and since then he has been in close contact with our institute and so it has visited us repeatedly and more recently has spent a longer period of time with a Martin Goods wheeler fellowship. I should also mention that the same happened again when I was in charge of Asia Pacific Center for theoretical physics in Poang in Korea because there's a problem was that one had to build up a respectful in-house research program which wasn't there and therefore again you have to try to attract the people who and to interest them in the center and to attract with their activity other people and Boris also organized there very nice workshop and with his friends some of them are here and that was also very helpful for us and at present he is helping Sergei Flach very much to set up an institute for theoretical studies within the IBS in Korea. So Boris thank you very much for everything and best wishes for the future. Now I would like to come to my talk and the problem is we have had already three talks on this topic started with Leonid's talk on Monday for his talk at all followed by a talk of Leo Kovanov and also the talk given by Daniel Ljos. So in today's talk of Neil Cooper also in the same direction so the question what is left so what I will try is to give it tutorial type of talk but even then I think some things are superfluous I live out. So so why are there of interest I think we have heard this several times I don't have to repeat it again people want to learn more about or realize my own fermions because they are topologically protected objects and what is important is you need superconductivity for that what's important is that there are only very few intrinsic topological superfluids examples of B phase of helium-3 and the copper-bithmers selenite so in most cases you will have to introduce superconductivity by proximity effect also what you should keep in mind there are always two possibilities to search for these Majorana particles either in a solid or in a system of ultra-cold atoms and the parameters are the relations between parameters in these two systems are quite different the three parameters always a fermi energy the strength of spin orbit coupling and the copper binding energy so in order to have a topological system you need a non zero charm number and also just to recapitulate in one minute so if you take any block state and you change this block state with respect to k and go on a closed path then the final state and the initial state will be the same except for a phase and that is the barry phase which can or cannot be zero or is not zero and this phase is given by the derivative of with respect to k of your block function and it acts like a vector potential so then if you convert this integral over the line in k space to an integral over s area then instead of the vector potential you have the curl of the vector potential and then this integral is something like a flux like magnetic flux and if you identify this area which you integrate with the bryon zone then you will get a result which is zero or not zero and if you do this for all the filled bands then what we end up with that is in the charm number and that has to be zero in order to have a topological system now and then if you have a topological system you have also had several times so then you have the edge modes and that makes a big difference but topological system is time of us invariant or not if it's not time of us invariant like quantum or effect you have edge chiral edge modes and they are classified according to their number and the corresponding object in the superconductor is again caramos but these are Majorana fermions and why because in that case you have particle hole symmetry and therefore when you look at the particle excitation in all excitation the same state so therefore you have only half of the number of states in superconductors then in topological for example topological insulator and if the system is time reversal invariant then the equivalent is the quantum spin all effect and they are the edge modes beer in pairs of two and the same correspondence you have in the superconductor these are helical states okay and he had said once more we have heard this several times in the previous talk so I can skip that he just indicates that whether you have a particle or whether it's the same state and therefore you can have a state which is pinned to zero energy we have also heard this morning in detail how you can generate superconductivity for example in a topological semiconductor or else even a matter you can do that by proximity effect and so that can be the one-dimension where you then have the Majorana modes at the end of this one-dimensional system or in two dimensions then it will be a vortex and this is done in order to have the vortex you have to put its topological insulator on top of magnetic insulator which generates a magnetic field and when you in when you inject superconductivity into these surface states here of the topological insulators then you will have edge states but those edge states will now not run only along the edge of the probe but also along the edge of such flux unit and therefore since the Majoranas always appear in pairs you will have here one of the Majoranas is just is associated with the flux line while the other is at the edge that should be a vortex not a vortex now I would really like to come to the superconductors with rush port types been orbit and action now if you have a single layer of lead deposited on a semiconductor then this is still superconducting and therefore in this system you have no longer inversion symmetry because when you have this layer on substrate then the potentials on the two sides of the layer will be different and therefore you will have here an electric field setup and then if you look into Landau-Lüftschitz what's the effect of an electric field it causes spin orbit in action and this spin orbit in action is one of the simple form indicated here so the spin is now always perpendicular to the momentum and everything is in the plane and the z is the axis perpendicular to this for example lead layer and then the you will have two bands which touch each other in form of diracone and there you see already what the whole thing with the direction things are driven if you have now a Fermi energy which is close to the cone you may have a topological superconductor so now these are two different surfaces which have different Fermi momentum but the Fermi velocity is the same both but the here in one case the spins are running clockwise they're always perpendicular to K in the other counterclockwise now if you have a superconductor now of this type then the question is what type of pairing do you have and you can have intra band pairing where you pair an electron here within one band and we'll pair this electron with the one here with K and minus K or you can have inter band pairing where you appear an electron with K with this band with minus K with this band but if you analyze this then you will find that intra band pairing has equal spin singlet and triplet contributions while inter band pairing has only spin triplet contributions therefore when you have a strong spin orbit coupling or better if you have a local a purely local interaction then only the intra band pairing will be left and assume now for a moment what would happen if the inter band pairing would be the important pairing so if you would have a particular type of interaction which favors inter band pairing so it could in this particular case not be a local interaction then you would have to pair always an electron far with K here and minus K here but since the Fermi momentum here are different in the two cases you see already that the spin orbit interaction will have sort of a pair breaking effect so spin orbit interaction will break the pairs and also if spin orbit interaction is sufficiently small so that's the pairs are not all broken then if if spin orbit induction is increasing in strengths you will eventually shift to a situation where you have a finite pairing momentum because you have now pairing between unbalanced populations so then there will be degeneracy with respect to the pairing momentum and that is a situation which very recently has been investigated by Yuri of Tinnikov now I will come back to this later because if you have a very strong magnetic field then it turns out that situation has changed then even for local interaction you will have the interband pairing will be the important one and not the interband pairing so now one question is what happens if the for example if you have such a low concentration low low Fermi energies that you in this regime would this change for example topological charm number anything like that because I also should have mentioned that in this case here this is not a topological system if you only look at this case here there's a charm number different from zero but here in this case you have also turn them up the opposite side so if they cancel each other so this is topologically trivial situation now the same holds true for this case here because if you look at this we have here this field region with the electrons so here outside here you have one charm number and here you have another which just cancels this one so it's topologically trivial now so let us assume that we have a large spin-off scattering so that we have only intraband pairing then as I have already mentioned before then the gap for loben is here see this diracone will be split and if we have then Fermi energy or chemical potential within this which is within this situation here then you have a topological superconductor because now you got rid of spin degrees of freedom to pair this electron with that so it's like you pair spinless fermions so this is in a situation that people hope that they can realize that with ultra cold atoms and they are now considerable activities in this respect okay so I could skip here now this is just once more in pictures what I've said before so I can go over that this I've we have seen in the talk given by Leonid can go with that so one problem is if you apply a magnetic field perpendicular to the surface that it has also destructive effect on superconductivity because you will have you will generate because of water says and that will the orbital effect of magnetic field will destroy superconductivity so what you really would like to have you would like to generate a topological superconductor which by applying a field which is within the play now this is possible as has been pointed out by alicea for very specific cases if you have a semiconductor of the what's that structure then which is growed in one one odd direction as pointed out in the talk by kufum normen and if you have at the same time a rush by interest laws spin orbit in the action then you can generate a gap by applying a magnetic field parallel to the plane in this particular case and this has been proven by alicea it is it can be seen that this is true by looking at a very special case namely when you have here the rush by spin orbit in action and you have here an anisotopic rush by parameter and if you assume that one of them is zero but that's remaining one is equal to the interaction parameter of trestle house see here in this particular case the trestle house spin orbit in action tries to align the spins perpendicular to the plane why the rush by spin orbit in the action tries to align the spins within the plane and this competition then enables situation where you can really open a gap by applying magnetic field within the one one one plane one one or plane now this brings me to the field to the special feature of applying magnetic fields within the plane and there we have we are dealing with a situation that we have here these two different circles so to speak two dimensional case with clockwise and counterclockwise spins with respect to k now when we apply magnetic field parallel to the plane then for some of fields applied in this direction then these states are favored they have a lower energy than the states at this end therefore we would like to fill those states here which implies that we shift the whole circle perpendicular to the direction of the field here on this surface just the opposite happens here these states are favored while those have a higher energy therefore we would like to shift the states again perpendicular to the magnetic field but in the opposite direction to here now what happens if we have intra band pairing so if you have intra band pairing with pair momentum zero then we pair here the electron here with k and then with respect to the to the old origin and minus k and here the same so this will be okay if the field is sufficiently small but if the field gets larger then it's more advantageous to shift the pairing together with the shift of the circle in other words we would like to pair electrons in the new rest frame so to speak with the opposite momentum so that is that pairing now the cooper pairs now acquire a final pairing momentum now because here and here we have then the same pairing momentum this gets very disadvantageous for this case because here in this case we would like to pair the electrons if we also shift them with the origin of the system so we would like to have a pairing momentum pair momentum is minus k over 2 so but this is if you would do that then we would lose a lot of phase space for the inter band pairing so what happens then in this case is that at a given size of the field the pairing momentum shifts from zero pairing momentum to finite pairing momentum and at the same time we have here unpaired states because here it is much better to move the electrons to the magnetic field according to their favorable spin direction to move them like this and leave the states on the opposite side as unpaired so here we have then as empty so here we have unpaired electrons and this will go on and on until with increasing field this whole thing becomes unpaired now when we have so eventually we expect then there will be a second phase transition so now what will happen in this particular regime when you partially unpaired electrons you will have then a spin current flowing so you will have a ground state of the system with a finite spin current so the electrical current of course has to be zero this cannot be no charge current according to Bloch's theorem but there will be a spin current so the spin current will flow in the direction perpendicular to the field and the direction of the magnetization will be parallel to the field so how can we understand that we will have here a ground state with a finite spin current now this in order to see this you have to consider set a magnetic field acting on a superconductor has always two effects one is on the orbits and the other is on the spin so we know if we can neglect the magnetic field effect on the orbit on the spins so if you only consider the effect on the orbits then if the field sufficiently high we will have at least a type 2 superconductor we will have an inhomogeneous solution namely the upper cause of vortices and in that case we have screening currents around all the vortices so we have a system where we have an electric charge current around the vortices and screening currents now here we have a situation where we have neglected the orbital effect and have just looked at the effect on the spins because if you have just a single lead layer with a parallel field there is no space for any orbital effects so therefore here we have a gain in analogy to the upper cause of vortices we have a gain in inhomogeneous superconducting state but instead of an electric current we have a spin current so that's the reason so then how would one be able to detect as such a situation that we have partially deep deep pairing now for example you would see this in the tunneling density of states with two-dimensional systems so here in this particular case these are numerical results for a given value of alpha where the this final pair momentum starts to set in you will then see filling of the gap and quite big changes in the tunneling density of states now what is expected is if the fields get larger even larger so in the limit of a very large field of the order of the spin orbit in action then we expect there should be a second phase transition then we eventually we expect that we will have then for one circle pairing this pair momentum q to q while for the other spin direction we expect a pair momentum minus 2q so they should be equal but opposite so but the details of this phase transition have not been really worked out but this is now done and numerically and for different parameter regimes so it's something which is worthwhile to check now then the question is what happens if you have two magnetic fields one perpendicular as we have said before in one parallel in that case and you will have the splitting of the two different pharmaceutical surfaces as pointed out before but now is a final time momentum because now here if you look if you look at this so if you are here in between this gap with your family energy then k if you pair an electron from here and here as this does not correspond anymore to k and minus k because of this distortion of these of these circles by the field in the plane as I pointed out to you before so from the very beginning you feel finer as you apply a field perpendicular to the plane in addition to the one I'm sorry by applying a field parallel to the plane in addition to the field which is perpendicular to the plane and you will have right away the finite cube pairing Cooper pairs with finite cube and you will have then a topological in homogene superconductive state okay so then the last question is what happens if you have here the situation where the field perpendicular to the plane is opening the gap here and the other field is now tilting the two minima which you have this now the field in the plane and if the concentration is so low that you're here then again you here you have gotten rid of spin decrease of freedom so it will be here pairing of polarized system and it will be a topological system but without spin current so this is a brief summary of what I wanted to say and so the conclusions are said it's that it's worthwhile to study more these ultra superconducting layers and also of course to study the optical atoms on optical lattices and to the it seems that there's still quite a number of interesting features which have not been really fully exploited and might lead to interesting new physics so thank you very much