 Now should work. Does it. Yeah. Okay, maybe not so good. Okay, so I will be talking about the two phases. And one is the spin singlet, a superconducting phase. The one is spin triplet. And I'll show that one is topologically trivial, while the other one has zero energy boundary modes with fractional lies spin quarter at the edges. Then I'll show that when this is in the absence of an external field, then I'll apply an external magnetic field at the edges, and I will show that actually even if I take a trivially topological phase. Then I have a, when I apply, when I apply external field. So when I apply an external field, I can turn it into a topological phase. So in other words, the topological nature of a phase is determined by the interplay between bulk and boundary. It's not a pure bulk property. It's a phase diagram as a function of the fields, and I'll show there is a structure of the Hilbert space, a something that occurs that is called a, I can state a phase transition with the whole Hilbert space changes as a function of this external field. As an interlude, I will talk about a system where instead of having external fields, we attach condo impurities at the edges, and I will show that three regimes occur. I'll go through that very quickly because I've given too many talks on that so I need something newer. Okay, then, so this, the superconductor is an example of what is called symmetry protected topological phase. And another classification, which the xxz the gap at xxz provides an example is what is called an SSB phase, a spontaneous symmetry breaking. Whenever I will show that the edge structure of both is very similar. So, very quickly, when we talk about one dimensional gap systems. There is a classification when they are without symmetries that topologically trivial, when they have a symmetries, then they fall either into the class of SPT systems which have unbroken symmetry. A short range quant a entanglement string order protected edge modes and degeneracies in entanglement spectrum for spontaneous symmetry breaking the typical landau type system we have broken symmetry again short range entangled entanglement in the quantum phase. So these are the two examples that I will study in quite detail. A quick comment. This is not the key tie of chain. The key tie of chain as we learned from said Rackian a in a very nice tutorial is a quadratic a superconductor P wave, and the superconductivity is induced. And from the outside we impose the superconductivity by a giving the expectation value delta to see dagger see a to see see and see dagger see dagger. And it's not that the model generates it internally through its interaction. So here is the model I'll be studying. So I'll call it an intrinsic or charge conserving for fermion interaction. The Hamiltonian has left and right moving fermions and the interact via for fermion interaction. So the right, right, and left, left, and they are two coupling constant G parallel and G perp. So it's the type of a you one a tearing model. A perp and G parallel are equal, then there is an enhanced symmetry and as you to very often called the gross never model. And I will solve this model on an open line with a open boundary conditions. So that means that if a right is moving reaching the edge it's reflected as a left. The left reaches the other edge. It's reflected as a right. So here is the phase diagram it goes back into the 80s. So, a, when G parallel is positive, then the model has to a fixed points. One is here and the other one is here. And the system flows to strong coupling where it generates a mass gap, or a superconducting or the parameter from anywhere it starts in this region a, or if it starts in a region, a dagger with G perp is negative, it flows to a dual a fixed point, which is So, a, as has been studied for many years. So the attractive interaction arching flows to strong coupling. The model has a very interesting poor property that charge and spin separate. That is what allows the model to generate a mass gap without a breaking symmetry, which is not allowed in one dimension. It has gapless columns in the charge sector, which is the couple from the spin sector, and we label the phases according to the dominant instability. In other words, the slowest falling correlation function in the phase as a spin singlet superconductor is this operator, and in this phase, it's the OSTS. All phases are related by duality, a Omega, which a flips the sign of G perp and allows us to map in a unitary form, the Hamiltonian with a G perp to a G minus G perp. So every point here is mapped to a point here, but the boundary conditions are not mapped. It's only the bulk part. The boundary conditions are, as I said, a reflection at the edge edges. And what I will show is that this phase is a spin. This spin singlet phase a has a non degenerate ground state, while that in the phase a while a dagger will have fractional charges. And these fractional charges are related to the fractionalization of Z two symmetry Z two symmetry is the spin flip symmetry up down, which is formally built by Cy L is mapped to toss it to Sigma X Cy L, which just flips the spin and R is mapped to R with a spin flip. And this model was with open boundary condition was more recently studied by Eris Berg and collaborators. We provided exact solution which we are going to allow us to study much further, and I will explain how. And as I said this is an example of an SPT phase. Let me first do a quick semi classical calculation. So if I bosonize the fermionic field, namely I express them in terms of bosonic field, theta and five dual field. So this is a standard formula that Hamiltonian be separates to charge and spin sector in the charge sector. It becomes a lot in your liquid. Well in the spin sector, it becomes a sine Gordon model with a cosine term, and the relation parameter in the sine Gordon, and the fermionic language are well known and studied and written down here. In terms of the bosonic fields, the number of particles is given by the integral over the charge field, while the spin is given by the integral over the spin field. The open boundary condition that were imposed on the fermions are translated in the bosonic language that Phi C at minus L is zero, while Phi C at plus L is given by square root of pi over to N, while the spin part is given again in terms of the total spin, and there's a difference between the left and the right, which determines the total spin of the system. And notice this look very unnatural, typically in the same Gordon people have studied of course with boundary conditions but rather with the replay boundary conditions. These are not the natural thing you would do in the bosonic language, but these are translated from the fermionic language, and they will be responsible to what a to the question. In the semi classical limit of this model when beta goes to zero while holding M not fixed. We are in this part of the phase diagram, and then a, the low lying excitations are controlled by minimizing the potential. So in the ground state configuration. I should have said a very important thing when we both on Isaac notice that there is a number here, hi, hi is plus minus one depending on the sign of G perp. So if it is plus one in the a phase and minus one in the a dagger in the a hat phase. Okay, so when in the trivial phase when Kai is positive, a, the ground state is of the has the values, a pi n is n times two pi over beta, which minimizes the cosine. While the if Kai is negative, it's n plus half a two pi over beta to minimize the minus cosine. It's easy to see that when Kai is negative. There is a mismatch between the values in the ground state, for example, this line and this line, and where we are locked to by the boundary conditions, which are this line, or this line or this line, which is that close to the edges, the configurations have to be all from the ground state value into the boundary value, and there are four ways of doing it. I can go from the ground state up or down on the right. Again on the left, either up or down. And I integrate now that change the spin in this configuration, let's say from minus L over two to deep into the bulk, I get a quarter, which is surprising plus minus quarter, plus for this minus for this. So at the edges, it looks like we have quarter modes. The spin has fractionalized and Kesselmann and Burke have verified this doing the energy. So how valid is this conclusion. So the really the calculation was done in the very anisotropic limit remember I took beta goes to zero in order to dominate the configuration semi classically, but really we want to have to study the model in the same with quantum fluctuations are enhanced. So, are the, are the results still valid. And what we did is to solve the model exactly using the beta ansatz, which allows us to get all the eigen states, including the ground state and the excitation, and ask whether a this results that we got and not a destroyed by fluctuation by long range fluctuations of the other modes, particularly that this model is an example, not of a gap SPT, but a gapless SPT, because in addition to the gap mode, we have a gapless mode, which decouples in the bulk but couples at the edges. So the beta ansatz, I will not give the details. When you solve it, you translate it to a set of algebraic equations, where the modes that you need to solve for are the lambda alphas which are called a spin rapidities or beta rapidities and the way you classify the system is by choices of integers. So what you have to find. You choose what you have is to identify what configurations of integers gives you the lowest energy. This is the ground state, and then by varying the integers from the ground set configuration, you can classify all excitations. So what I will show is that the boundary. So here I have the bulk term, and I have the boundary term. And I will show the fact that the boundary term has here this integer chi, which can be one or minus one leads to a dramatic change in the ground state configuration. So solving the equation solving the equations. So solving the equation in the phase a where the spin singlet superconductivity, we find a unique ground state and the singlet. Whenever solving it in the spin triplet a phase with a superconducting phase, we find that there are four phases. There is an s z equals to plus minus half. And that is when all these parameters are real lambda real. So we find two more solutions which are singlet, which are associated with edge modes. And one, one edge mode is when lambda is equal plus minus half, and another one when it's a shifted. What we find is that they in the triplet phase, we have to degenerate ground state, we spin plus minus half, and two singlets. And we interpret the states as a rising from edge King state, which are fractional spin a z plus minus quarter, plus minus quarter on the left, plus minus quarter on the right. So we assume that there is an operator acting an edge genuine quantum operators, whose eigenstates there are something that I will re examine in a minute. So the class if the Hilbert space of these edge Kings is now classified by the total spin left and right. And the two states with z equals to zero are generated by quarter minus quarter, or minus quarter quarter. So quarter minus quarter, or the other way around. And the two states with s z a plus minus half our quarter quarter minus quarter minus quarter. So this is a DMR G a calculation, which verifies that we have here a spin density localized at the edges. And fits very well with the quantum calculation. But we have to ask whether these are indeed sharp quantum objects and there is corresponding quantum operators so I can values that are, or only spin accumulations at the edges. The typical way of doing it is to study the variance, namely whether s z minus a quarter square, which can be calculated via DMR G is sharp and going to zero as L goes to infinity, which was carried out by the our group and indeed it satisfies this criterion. So we can indeed a argue that these are indeed quantum sharp objects at the edges. So, to make sure that this is indeed an SPT phase, a, ask them to calculate the degeneracies in the entanglement spectrum which come out a beautifully. And you never to our surprise. Also, when you do the same type of calculation in the trivial phase you find degeneracies. So, this is probably not a very sharp criterion, and you have to resort to more checks. Okay, now I'm coming to the main part, the main half of the first part. Yes, it's exponential with the size. Yes, the question was whether is a, the difference between them goes to zero with the size exponentially or power light. And it goes exponentially. That is what the variance essentially checks. Yes. There is correction coming from a the charging energy which is of one over L, one over L correction, but that comes from the charge sector, because there is a gap less sector, which are the charge the whole loans, which the couple. Sorry. Exactly. So now we asked the following question. What happens if we apply a magnetic field at the edges. What I want to argue is the, what was a trivial phase, namely, the spin singlet was trivial, when we had an open boundary conditions. Now, when we are going to apply magnetic field at the edges, and I will argue that it becomes topological. So in other words, whether a system is topological or not topological is not only a bulk, at least in one dimension, but a combined a effect of bulk and boundary. But they go to in, I take the infinite volume limit. It matters, but in a trivial way. Now, in, of course, when you do the MRG, you see the effects very clearly. But in our calculation when you take a, in other words, it would go into the quantum number and that I calculate, for example, which is the integral of the Latin jar liquid phase, but not in the spin bulk, where the superconductivity occurs. Yeah. But in a controllable. Exactly. Okay, so the model applies a magnetic field or twist fields, as I've written here, the model is still remains integrable, and we can control it in a very precise way. Remember that the model has the Z two symmetry under the Z two symmetry. The boundary condition the boundary fields are map be left and right goes to Sigma X, which is the spin fleet be Sigma X, which so the parameter. The epsilon that a here, here is epsilon L which characterized the left side. Here is epsilon R which characterized the right side. A, the mapping the spin fleet mapping corresponds to epsilon goes to minus epsilon. So what, as a result, you see there are only two values of epsilon, which are consistent with the Z two symmetry, either epsilon is zero, and this corresponds to the open boundary condition, and the, there are no fields, or epsilon is equal to pi over two, which corresponds to epsilon being a fixed point that pi over two, and this corresponds to a field acting on it. So, this corresponds to this field acting on it. So, what we see then that by applying a field I can flip the sides. The two values zero and pi over two are a eigenvalues a of a Z two, and they are R G invariant parameters. And for these values, indeed, a day edges have zero energy. But if I move away from those values, then a, the epsilon prime measures the amount of moving away. So we will have now a not zero edge modes, but meet gap modes, namely the edge modes will acquire energy, but it will be below the mass gap. So this deviation, the edge modes are no longer symmetry protected and no longer zero energy modes. Remember, I have this duality symmetry, which maps one face to the other. So, before I was here with open boundary condition, the blue was the trivial phase, and the red here with G perp negative was the topological phase. If I act with the duality transformation, the topolog, now the spin singlet phase becomes topological, while the spin triplet becomes trivial. So we can map phases back and forth. And what I will now study is the deviation in the spin singlet phase with open boundary with the dual open boundary conditions, and I will study how they deviate from the zero values. So what let me look at the face diagram. Here I apply magnetic field on the right edge modes on the left edge modes. So in the absence of a magnetic field, the model is still topological, and the phases are, and the energies are zero. When I begin to move away. Edge modes acquire energy. And then when they reach a critical value, they melt into the bulk. In other words, the value reaches the value M, and they become part of the bulk spectrum and disappear from the edge. And as a result, I have this phase diagram, and the number of mid gap states in the region A, for example, here is three, then it is in the B, it is two, and in C, both sides have melted into, have leaked into the continuum and disappeared from the spectrum. So I can analyze now each region separately. Let me think about the a region where the epsilon primes are less than one. So I have a, I have spin half in each case, which is below the mass gap. In this case, what we find is that I have only the minus one half excite a ground state remains gapless, while the singlet singlet prime and the plus half become gap. And the energies by which they become gap is given here by this expression that M left and M right are given by the mass gap time, sine pi over two epsilon which gives the deviation. So you see when it reaches pi, it becomes equals to M and leaks into the bulk. For epsilon less than one, it's going to be a mid gap state. So the ground state a of the minus half. So now the edge state split the minus half still remains with a zero on the right I'll have, if I have epsilon R and epsilon L, I'll have MR and ML, while the class half now has combined energy, and only the minus half remains a zero energy gap. And we can associate to this now a operators that create and annihilate the particles. So for example if I start with the ground state, which is minus half, and add a fermion to the left, it will not go into the bulk, because it has a lower energy zero mode. So if side dagger up added to the left will turn the state into a singlet. Similarly here side dagger up acting on the left side will turn into a singlet, or if I add particles to the left and to the right. And a plus half a state at the edges. So, I can reproduce these processes by introducing creation operators a dagger to the left and right, which are acting on the ground states minus half, or a and creating the various processes that I described, and the effective function of the Hamiltonian, then is very simple. It is just a quadratic Hamiltonian with ML depending on the deviation, or on the imposition of the gate truth. So as a result, we'll have various towers in the Hilbert space will have the towers that we built. For example, let me describe it here. So here I have a tower of states built on the ground state. Here I have tower of states built on MR. The tower is below the mass gap or on ML, or on ML plus MR. Now, if I keep changing the magnetic field, these will leak into the continuum, and will disappear, and then I will have only one tower in the C phase. Here's an example of what is called an eigenstate phase transition where the full spectrum, typically when we talk about phase transition, we think about the ground state the change in the ground state and the low lying eigenvalues around the ground state. So if we talk about a eigenstate phase transition, we examine how the whole structure, the structure of towers built over the various a eigen modes, how it changes, and we have a very interesting phase transition taking place as a function of these external fields. So let's do this and go to the interlude. So, I wanted now to think about a model where instead of having external fields. I have now a quantum upper a quantum spin. So this is nothing but a condo spin, coupled to a superconductor. Again the model is integrable. The kinetic energy left right moment. And this is the interaction term and this is the coupling to the spin. And again, we have here open boundary conditions. We tried also to solve this model, not this open boundary condition but put your magnetic fields but we have not been successful thus far. So, but instead of having a magnetic field now we have a fluctuating quantum spin have a condo. And what we are studying here is the condo effect in the presence of superconductivity in one dimension. And the model is integrable for any coupling J, which is the condo coupling and coupling G, which generates the superconductivity in the bulk. And as you can imagine, there is going to be a competition between the two types of non perturbative effect. If the case very large compared to the gap in the superconductor will have a condo phase, if they are of the same order of magnitude. Then we'll have what is called the YSR or sheba phase. And if the case very small compared to Delta, then we'll get an unscreen phase. Actually, I'm lying here a little bit, because TK is generated only in this regime in the condo regime in the sheba phase. And in the unscreen phase there is no condo effect and no TK. So if then, instead, what we have is this three different phases here in the phase which is the condo phase, J is very large compared to two J, a renormalized condo effect takes place. It's not only the condo but it's renormalized by the quantum in the bulk and TK takes different values in the YSR phase. What happens is that the condo impurity binds an electron to it, and in this part it is screened. And here there is a first order phase transition where it becomes unscreened and connects to a fully quantum unscreened phase if J is very small compared to two G. And the quantity that determines where the phases are located is a renormalization group invariant combination of these couplings. Remember, all these couplings run under RG. J and G are not a RG invariant when we change the cutoff they run, but this combination D that I've written down here in terms of J and B, where B is this combination is RG invariant. And when we have, and we alternatively use D or A depending whether it is real or imaginary, if it is real it's D, if it's imaginary I just pull out an I and call it A, then this diagram tells us where the phases are located. We can calculate all very nicely all the properties density of states and a magnetization in the various phases. We also have two supercondate to impurities at either edge. And so, up here, a one here and one here. Now the model is integrable in with three independent parameters, G, which tells me about the strengths of superconductivity. And J, let J right and J left, which I combine into two are G invariant AL and AR, which characterized the interaction on the left and on the right. And again, I get a very interesting phase diagram. In particular, I didn't finish. Okay, in particular, a, what we find that we don't understand why that if AR and AL are equal to one, we get a supersymmetric algebra. Whatever that means. Okay, part two in three minutes. Okay, so when you do what you find also very similar structure in the symmetry broken phase of the X, X, Z model. And my tough chairman demands that I stop for questions. So any questions. Okay, I'll go to the end. Conclusions. So we studied the intrinsic charge conserving superconductor, we identified topological phases. We can show that we can induce the topological phase by applying boundary fields. And we found a plethora of mid gap phases. We discussed the condo superconducting combined system. Again, we identified a, it's interesting because in three dimension if you study a, the condo superconductivity system with BCS, you get only the YSR phase. Here you have full three phases as opposed to the situation. And then we got a phase diagram of fractionalized edge modes in the Heisenberg chain, and various boundary aspects which are common to spontaneous symmetry broken system. And then we got a SPT symmetry protected, which are different classification. So there are a lot of things to do for example, if we generalize it to SUN, instead of Majorana edges we get, we expect para fermionic edge modes, which would be very interesting to store and a information in. And we can couple interesting question is how to couple wires at the edges and propagate the information. And there are some other questions that I can explain now. Okay, thank you very much. Okay, thanks for the great talk. Maybe, if you can explain better more about the shipper versus non shipper. So if I, if I just think of different phases by the, the difference in the magnetic in the susceptibility is respect to the local magnetic field, suppose a couple fields to a local spin. Yes. And ask about whether it's clear wise or power, or what is. Yeah, so, so the susceptibility. Yeah, changes from. I appreciate it, but it changes once or twice. It takes changes continuously here crosses over to this and it changes everywhere. So for 3d. Okay, I mean, if there's a big activity in some sense there is no content because, well, as you know, but doesn't mean that basically the local spin maybe not screened. Right. Well, it's or screened. Yeah, and then it just explicit was a particle that goes to infinity. That's right. Okay, so there is basically one transition when it crank up interaction strings. Yeah, right. So now you're saying that here is not one transition. So, these are not phase transition these are smooth. Okay, so it's still one except at this point. So is there still one phase transition. Yes, from curious wise. Yes, within the wise our face. There is a first order. I think we call it whatever but I mean, yeah, it's one position from your vice to power. Yes. Okay, thanks. Okay, we have time for one more question. These towers of states you're talking about before the two phases. So, are those states protected in some way or yes, for example stable against perturbations that would break interoperability. It's a very good question. And the answer is yes, we examined this question for the exit Z where we added next nearest neighbor which breaks integrability, and it's stable. Now, of course, depends how violently you break the symmetry, but not not into into integrability per se gives you access to it but it's not the reason that it's. Is there any questions online. Okay, so let's thank