 So, first of all, I would like to thank the organizers for having me here, and yes, no problem. So, as I was saying, I would like to thank the organizers for having me here. So what I would tell you about in the next half an hour is a recent work of ours that appeared here and would soon be on PRX about, say, using ultra-cooled atoms not much as a quantum simulator, but rather as a way of creating states of matter or exploring states of matter that might be hardly accessible with real materials. And that's actually another conjugation of the whole idea of quantum simulation, so the one of bringing, say, ideal models into reality. So, yeah, so the underlying motivation is that topological phases of matter are at the focus of the interest in the last decade at least, if not more, but for academic and practical interests. And what I mean, practical interest is that there are people studying how to exploit spin currents or other kind of topological excitations for the next generation of memories, even if they are not quantum memories. So, but, say, I hope you had some kind of introduction about that last week, but we will explore what we need in the beginning of the talk. So and there are two open issues in this field. One is how to convert, say, paradigmatic models that are, like, very neat and on the blackboard give you clear topological signatures into real materials. And some key names are Britannia, and one of them was mentioned yesterday about the Arper of Stetter model, so this square lattice with fluxes. So how to convert or what is the relation between the paradigm model and the real materials is something that is a burning question around. So there are studies showing that some compounds that are called irridates might be an incarnation of the Kittayavonic model and so on, but of course they are not as tunable as a quantum platform based on cold atoms could be. On the other side, the role of interaction is often neglected in real materials, but it might lead to new correlation effect or to new phases of matter. So in order to tackle these two kind of questions, so my plan or my interest is to do that by using a combination of synthetic quantum methods, so for example ultra cold atoms, so the flexibility that you have in designing the Hamiltonian, tuning the interactions, putting gauge fields and so on, you have seen all of that last week introduced by Fabrice, I guess. And a combination of that with say a quantum information twist in the approach, so numerics based on quantum information, so namely tensor network algorithms, and as far as I know there was also introduction by Norbert about that. So let's get started, so let's get directly on the model and we will learn the concepts along the way. So the model I want to consider is a so-called Kreuzladder, so you can look at it as two internal degrees of freedom of some atom sitting on a site of an optical lattice, so horizontally it's the site in the optical lattice, vertically it's the internal degree of freedom, you can think about an hyperfine state. And then you want to have two ingredients at first, so you want to have a spin dependent complex tunneling, which is the tunneling with an amplitude plus IT in one direction for the upper leg and minus IT on the other leg. Plus you want to have a real spin flipping tunneling between the neighboring sites. And if you sit down and do the Fourier transform of that, you will soon realize that this is a model that describes two flat bends, I mean two because the unit cell has two sides and they are flat in the case the two intensities are equal to each other. And a way to see that is actually to look indeed in the Fourier transform and see that you can write the Hamiltonian as a pseudo-magnetic field times the set of Pauli matrices and the pseudo-magnetic field has the same amplitude everywhere and it's rotating once around the origin and say the same amplitude tells us that the bends are flat because the energy is essentially the distance from the origin and the fact that it's wrapping around or winding around is telling us that the bends have also a topological character. So that's all what we need to know about topology here. Now, well, if you are wondering whether this is at all possible to start with in cold atoms, there have been experiments measuring this Zuck phase in wire arms interferometry. So now what I want to do still at the single particle level is to add Zeeman splitting between the two the two letters and if you do that you will realize that you bend the bends sorry for the twist and they will touch at some point when the intensity of the Zeeman splitting is equal to the bend width so and they will touch and they will form an undouble direct point and that's actually the original motivation why Croix introduced the model in order to avoid fermion doubling in high energy lattice gauge theory simulations. Okay, that's still so you can see it also in this picture of the circle of the pseudo magnetic field you will see that this circle is moving away from the origin and at some point is barely touching the origin and this point in which it's barely touching the origin corresponds to direct point and if you go farther with the detuning you will have the circle that is not any more topologically equivalent to the one before. So you will reopen a gap and get bends that are not topological but that's a well understood it's all single particle physics and for the ones that are interested in details I only stress that this particular incarnation of the model has only a chiral symmetry so it falls into into the class A3 for the one that are experts on topological insulator classification. So there is yet another way before going to many body physics so to interactions and to actually the interesting part. So let's have a look at how can we understand these flat bends in another language and these flat bends can be understood in terms of a basis of localized states which are called Aranov bon cages and the language comes from the old community of Josephson junction arrays back 20 years now and you realize that you have these such cages because actually you have interferometers so if you look at the pattern of phases you will realize that from one point you can always find two parts that have the same optical length so the same number of openings that are self interfering and therefore if you start with a wave packet in some place you will always get not farther than a couple of openings so you are trapped so you have localized states and indeed so you see here the two cages that are present in the bark and they correspond to the two bends. Moreover the fact that the bends are topological tells also us that there should be some zero energy mid-gap edge states and that these edge states will have indeed say well okay so zero is an approximation as soon as we put this unbalanced they would be exponentially close to zero. But still so you see the Hamiltonian I can rewrite it in this way and this will be useful in the following so we will need this rewriting in the following. And let me only stress before going on that this model has been the workhorse for a lot of other studies of topology, exotic super fluidity and so on. And if you are wondering about these edge states also for that well there is not a measurement yet that I know but there are proposals how to measure it by brach scattering techniques in cold atom so it's also verifiable. Now the twist so the thing that we want to explore is what happens if we also add upward interaction okay so if we add interaction well okay you can think of that as u I mean we picked up a strange notation but that's the standard u and that's an interaction for particle sitting on the same physical side with spin up and spin down. Now the question is what happens to the phase diagram so along this line of non-interacting we know that we go from topological to non-topological but what happens elsewhere. And since then the details might drift us away from the old picture I prefer to give the old picture in the beginning and then reanalyze it in the following. So in the old picture is like that so to sum out to our surprise we found out that not only well so you have like a transition between the topological insulator and the trivial phase that gets bent and reanormalized but that's kind of standard but at some point if you move instead in the direction of not having Zeeman imbalance or having a very little Zeeman imbalance and you move the interaction strong enough then you get into another topologically trivial phase but that has a different order character with respect to the one above so the symbols stay for ferromagnet and paramagnet and topological insulator it will become clearer in the following. And for the ones that are interested in the field theory behind this model so we can also predict that this line here is a c equal one line so it starts as a c equal one line which is expected because here it's a real fermion it's a full fermion that is getting gapless and then opened again but at some point it splits so at the triple point it splits into two c equal one half lines and this is not only out of numerical evidence but we have say analytical ideas how to justify it and indeed the plan of attack which is the outline of the following in the talk is to say employ analytics and map the problem onto different ising model onto different effective ising models in order to see that this transition is indeed an ising transition as we find numerically this one as well and this one is actually not a xy but these are or any other u1 transition but it's two ising transition that are sitting on top of each other they are like going parallel so this c equal one is coming from two c equal one half that sit on top of each other. Say that's the analytics then say we employ numerics based on matrix states as I was promising and then we do say some little bit of entanglement analysis entanglement spectrum and entanglement entropy analysis and then I will come to the to the a scheme how to realize the model or to explore the model in experiments. So okay so let's get started from from this side of the phase diagram so at zero interaction we know the picture let's see what happens at weak interactions at weak interactions the first thing that you want to look at is whether the whether the edge states survive and to see that you simply you can simply look at the compressibility gap versus the the the genera C split so you can compute the energy to add or remove one particle or the energy of adding two three or a few in the thermodynamic limit so if the the insulator is a standard insulator these two quantities should coincide in the thermodynamic limit and indeed that's what happens here so different different colors are different sides of the systems and say the dashed line is the the genera C split versus the the continuous line is the is the compressibility gap and you see that they collapse to the same line on the other side in the topological insulator phase if you do the same if you perform the same finite size scaling analysis you will find out that the the the Genesis split collapses to zero whereas the compressibility gap stays finite so the phase is gap but there are two degenerate ground states no it's it's correct so a little delta goes to zero so that's an indirect so or say that's an energetic way of looking at whether the the edge states are still there but we can look at another another indicator and the other indicator is well we are looking at this transition it means that we are looking at driving delta epsilon so Zeeman splitting so we can look at the observable that is coupled to that which is the imbalance between the two legs and if we look at the imbalance between the two legs we see that this is say roughly a magnetization and it looks very much like an icing magnetization so the behavior looks very much like an icing magnetization and indeed if you if you compute its susceptibility and you scale it it all perfectly nicely fit with icing and that was puzzling at the beginning until we realize that that indeed there is a mapping in which you have to do a little bit of algebra and which you have to do say a bogolyubov transformation followed by Jordan Wigner one and this bogolyubov transformation is mixing the upper and lower leg so it's a bit non-standard but if you perform that you will realize that that zero interaction you have two perfect copies independent copies of an icing model and that's justifying the fact that the magnetization here is is looking like the one of an icing and it's also justifying the following fact that that it will be c equal one okay and as soon as you put interactions the coupling between these two icing models is magnetization to magnetization and then you can try to do self consistent mean field and get say perturbatively some some expression for the transition line and that's the red line here and it fits very nicely with the data okay but then you can think okay this will be a line and then it quantitatively deviates from self consistency in field but it's like kind of boring it will extend up to infinite and finished actually not because say or at the beginning you can have this impression because you go to strong interactions and at very strong interactions you can also do say an effective model it's kind of a standard machinery so you project on the singly occupied ranks because you have a very strong interaction so you don't want to have two particles sitting on the same site right so you have only one particle per site and then you have the degree of freedom of being in the upper leg or in the lower leg and these you can describe this as a spin one half right so if you do that and you compute the super exchange coupling which is the same thing that you do in the upper model to get out antiferous magnetism you will get an effective ising model described by these spin one half operators that are called T and you realize that TZ is the same that we were using before so it's the number of particles in the upper leg minus the number of particles in the lower leg so it's the one that you intuitively would call Z spin and why is the proper the proper conjugate thing so the effective model looks like that and indeed if you look at the proper observables here so at the susceptibility of the magnetization in Z direction you get again a nice ising scaling if you look at the why so at the peak of the structure factor of the why why correlations you get like an order parameter that is nicely scaling to finite on one side and zero on the other describing this transition so here then so at very strong interaction we have a transition from something into something where one phase is ferromagnetic in this language and the other one is paramagnetic and now at first sight you will think okay that's only the continuation of that line but then you realize oh here I have a single ising model right so I have C equal one also something else should happen somewhere else and moreover this something else should happen some year in the middle because if you compute the the order parameter for for this phase here so this TY TY correlations and you compute it deep in the in the topological insulator this is zero okay so again there should be another transition here between these two so in order to see which kind of transition now we need this language of the of the cages right so we mentioned at the beginning we can write the Hamiltonian either in the real space basis with where the creation and annihilation operators are are the one creating and annihilating fermions on single sides but we can also change unitary lead the basis to the one of I don't know bomb cages they are also localized base so you can read the peaked the thing as a ladder where now the sides are lowest cage upper cage so lowest band upper band and if you work out the mathematics you will realize that in this language you get an an extended and quite exotic upper model so you get like nearest neighbor interaction pair per tunneling density tunneling a lot of other terms that people are usually looking for without so with with long range interactions and so on but here you get them for free and a crucial disclaimer or say a crucial annotation for the for the ones interested in similar calculations we are keeping both bands at a difference with with most of the literature so we are not doing a projection on the lowest band because otherwise you miss the symmetry of the of the transition actually okay so and something that I will not talk about but if you are interested you can look it up in the in the paper is that in terms of these of these extended upper model you can also reformulate the full the full description of the face diagram in terms of a funnel under some problems of an impurity problem where your age states are the impurities and they decouple in so they couple into the the bile can get hybridized and get washed out okay so it's weak in balance indeed so we can now define again and either the ising model where now we are restricting to the single single Aron of Bay bomb cage manifold so so you are restricting to the fact that either you are populating the lowest band or the upper band for each of the models and you derive an effective model that looks like that that is again an ising model where okay now the the the thing that is nearest neighbor interacting is Tx and there is another Tz and this T t tilde Z is now written in terms of the cages okay so it's not anymore written in terms of the local so the interpretation is a bit complicated but it should be very much related to the current that is circulating around the placket so either it's it's going say clockwise or anti clockwise roughly speaking okay and again you have a single ising model you get a single ising model you can predict from from that where the transition should be and that's actually where where you also find it numerically at this interaction over tunneling equal to 8 and again here you see that the thing is fitting nicely and now one interesting annotation is that if you would perform a Jordan Wigner transformation on on these ising model you would get for the bogeyubo of modes some effective bands which are now not anymore free fermions in the original language because here you have interactions and so on it's some some complicated mapping and remapping but you get some effective bands that are also starting flat getting bent down touching dirac like and reopening trivial but this time they are say these effective bands are flat in the strong interaction regime and they're getting trivial the other way around but there are like a lot of non-local mapping so there is no contradiction in it but it's it's still interesting to see that that you have say kind of the same picture of flat bands getting deformed and touching and so on dual between the two and it's actually something that we would like to to understand better and in particular we would like to understand better so feedbacks are welcome if somebody has ideas what exactly is happening here at the triple point and why this line is so seems very fine tuned for a lot of for a wide range of interactions and then at some point it departs well okay that's what I was mentioning before about the impurity model in a nutshell if you would write the impurity model which we have not done here you will connect it to the topological character in the sense that you will get edge-edge interactions mediated by the bulk and this will shift energies but not least the degeneracy but you can also look at the imaginary part of the of the coupling and look at the phasing and you will get the phasing as soon as these Bogolyubov modes get gap so as soon as they get gapless which means as soon as these strange very curious effective bands are touching each other okay and this again explains this transition here in terms of the edge modes get washed out yes okay and then the numerics most of the numerics that you have seen that was complementing the the analytics was done by matrix product states I guess that you all know this this concept of having many body ill birth space that is extremely expensive but thanks to the area law actually the picture is not so dramatic so that the physically accessible states are a very tiny corner and therefore you can do an economic say quite cheap description of your many body states by by tensor network the compositions and most famous is the variational de-emergy approach but there are several others and there are plenty of different the composition in terms of products but that's something that does not affect us here too much but what can we read out of this matrix for state simulations well naturally we have access to the entanglement spectrum and to the entanglement entropy in particular which is roughly speaking you bipartite your system in in two parts and then you look at the the entropy of the matrix of one of one one part of the of the system and according to conformal field theories studies you can connect directly the way these entropy scales with the sides of your of your block with the so-called central charge and here you neatly see I hope that the colors are visible so these are three sample points but we have done it for all the points that are depicted here along the transition lines that along this line here that we were predicting two couple dicing models and therefore twice one half so central charge equal one you get this blue scaling here okay and the only fitting parameter there it simply is non-universal part this is non-universal shift okay while if you look at these other transitions so to the topological to ferromagnet or to ferromagnet to paramagnet you will get these other two lines okay and indeed they are say same slope but different non-universal part but that doesn't matter okay but actually you can get out more and what what is that more that you can get out is the full entanglement spectrum and you can look at the lowest part of the entanglement spectrum say log of these eigenvalues and you realize that in the topological insulator phase so these are random points inside the three phases that in the topological insulator phase you will you will have a tower that is always doubly degenerate in your entanglement levels whereas in the paramagnet and in the ferromagnet you get say not specific patterns of the generation and this is an indication that this phase which is known to be say asymmetry protected topological phase at zero interaction is somehow robust also to interactions in so you can extend the usual paradigm of looking at the entanglement spectrum also there so it together with the presence of the edge states that we measured by energy and so on it's all indications that this is a topological phase overall okay and I guess that I have one minute or even less so let me only flesh the experimental ingredients and the idea is to use a lot of of the ingredients that probably Fabrice have introduced introduced last week which is so use a state independent gradient in order to suppress the spontaneous tunneling in your optical lattice and and then we want to use a weekly spin dependent optical lattice in addition that is that gets modulated in time and by doing that you can well you can work out the fact that that your effective tunneling matrix will we'll get a vessel function and not only a vessel function but if you play enough with the phases of this shaking you will also get phases so you will realize the plus I for the upper leg and the minus I for the lower leg and on the other side you want also the spin flips but the spin flips you can get them by standard with quote marks Raman assisted tunneling only with three different frequencies that are properly tuned and here it's a spot for our old papers of ours with another possible implementation that does not rely on shaking but only on energy selection rules but forget about that for the moment we can discuss details and so this is the overall picture and perspective so overall we have found say this we have explored this this model which is a traditional workhorse for one dimensional or quasi one-dimensional topological phases and we have found that under interactions you get a splitting of a of a direct line so called direct line C equal one into two Majorana ones we have not yet explored what does it mean for the topological excitation so whether we are mistaken and somehow the phase down there and we are calling for a magnet is somehow instead also still having some kind of fractional topological excitation then we have I did not present it but we have done so we can look it up mapping some two impurity problems and broadening of edge modes and so on so you have a new twist to understand the topological transitions well we think that we have high feasibility and detectability in in current or say next future experimental setup so I hope that somebody will realize it and there are a number of questions about what happens if we have now non-perfect flux so if you don't have pie so if you don't have perfectly flat bends and what happens away from our feeling because all of that was exactly at our feeling but then as soon as you go to fractional feelings then a lot of other things can happen say emergent symmetries fractional effects and so on and I leave you with a list of other recent works that are related somehow to exploring to the same kind of red line which is exploring topological or interesting phases with ultra-cooled atoms by a gauge fields and so on and I thank you for your attention and I forgot to put the banner we thank you for your attention