 Okay, can you hear me? I think normally I'm quite loud, so okay first of all indeed I would like to thank the organizer for the kind invitation and assembling this nice cool and worship so just to set the framework so this is the title of my talk so will be again quantum simulation preferably with ultra-cold atoms but okay many of this idea can be applied also to other systems for example photons and I'll I see the school I will try to to introduce briefly what does it mean this synthetic of studies trip or synthetic lattice and started really from from the beginning so the plan would be just few words about integer quantum old system although we have a very nice tool by Fabrice and also by by Michael on topological aspect then I will go to a possible realization of this kind of system that is especially suitable for seeing edge state that is this synthetic lattice they use this idea of extra dimension so using internal degree of freedom as site of an additional auxiliary dimension and then I will I will jump to a new result and we will show that this narrow system not only have a V state but have also some memory of the topology of the bike although seem paradoxical because the bike can be reduced even to just one site and then from this I will go to the interaction because this kind of system are very promising for considering interacting system and although they are very deduced they have already some interesting feature that have some connection with the larger system and I will go to review a bit this idea of my this analogy of my nervous phase transition in ladder and then while we will explain to you what we are working on now that is to see what is the effort of demeritization in this kind of system and we have quite some starting question some nice effect okay yes I told you this is so all all effect we know just Lawrence force quantum wall is much more interesting because it was quite surprising to observe this precise quantize conductance and okay this there are still some mystery of the when there are interaction with the fractional plateau but the integer one I will understood here we will focus mainly on the lattice case that is in the one discussed by by Fabrice and essentially we see that the effort of the magnet of the magnetic field reduce really in having this complex tunneling that is just due to the fact that we have a magnetic field that induce some I don't know boom effect while this this model is so nice because somehow it's a paradigmatic example of the topological insulator so in a rough world that is a topological insulator is a system that would be an insulator in the back but this conducting to the edge so how we can understand it so semi-classically if we have a very strong magnetic fields that is implemented by the flux the electron would be just confined in as close to control orbit so this justify the fact that they are the system become insulating but actually there are some surprise when we go at the quantum level because first of all we see that appear this very nice fractal structure and this is associated to topological property of the system this means that for each of these band when there is the gap we can associate a topological quantity that is the chair number and we had the Michael introduces me this talk but essentially roughly speaking the chair number is like now having curvature in the in the brilliant zone so having curvature in momentum a magnetic flux in momentum and this imply that okay when we put now if we put some boundary in our system even semi-classically we understand that this cyclotron orbit when they eat the the the boundary cannot close the orbit so we have this keeping orbit that produce some current but now due to the the presence of the of the chart this current is quantized and actually this is the what is responsible of this quantized condensants and what are this state that appear here that are confined here and the robust are exactly the search states so we want to see the search state because they are nice and interesting so we we know that okay we needed this magnetic flash the lattice and also a short boundary okay so as Fabrice say that this is obvious we can do optical lattice very well and we can load the BC for instance on the journal gas in our favorite lattice now there is this problem of overcome in the sense the atom are neutral but we want to have them behaving as charge in magnetic in a magnetic field so what we need is to engineer some synthetic current of boom effect so they should acquire some flash some phase when they tunnel and as Fabrice said there are very ingenious way for for doing it and they are will just by reason of time I can just confine to to one specific one that is this one that use this ingredient extra dimension plus lima laser somehow can be seen as an evolution of this seminal proposal by just can solar of about 15 years ago okay so first of all extra dimension very briefly so we have seen in also in Christian talk that we can have easily one d to d three lattice and relies upon model the one also by look and in this talk and I say now the question here we post some time ago was okay can we do for D now can we do it for the lattice in principle one would say no but actually dimensionality in the lattice is connectivity so what we need really to provide the the sufficient number on neighbor so to understand this how we can do suppose that this is our I represent with this cubic lattice genetic hypercube one so you see you have this hypercube you can always mentally think of distinguish the layer in the plane in this case the different color but now equivalently you may think that you have a system that lives in the plane where but there are some internal degree of freedom some hyperfine state of the atom spin state but also other stuff so getting the you get the back the the original system you're available just to couple coherently this this internal degree of freedom so obviously this is by far not restricted to speed state of the atom can be anything can be even momentum state of the atom trap mod in the trap but also not can be not just atoms called molecule photonic crystal and the list is very huge I cannot put even reference because it is not fitting so this was was very nice but okay after some time and already this is quite some time ago we realize that the same system could be used to get to realize easily this this quantum all strip we and the soft other steep we're saying at the beginning so now it's even more basic suppose you have some atom that are some internal degree of freedom and you have you put a spin independent what the spin independent lattice so if you do nothing you have just three copy in this case of one the lattice quite boring but now if you shine with some Raman Raman coupling that is just a Raviose elation between the states you see that automatically you induce the tunneling in this extra dimension of synthetic dimension and this tunneling at the same time is complex due to the momentum kick transfer by the Raman and now this is exactly what you realize immediately that this kick is depends on the projection of the difference of the k vector of distillation along along the lattice and so it's it's equivalent to have a constant flux so very seem very easily one almost for free one is achieving this constant magnetic flux and also just because we have a finite number of internal state we achieve a sharp boundaries so it's ideal system to observe a state okay so now briefly how this we can really speak of edge state how this edge state come around so this side just three three side they can go can change cage I can go and get translational invariant and they are going to lies at three by three Hamiltonian so I can speak of a state but if you look if I'm in the limit in which this transversal coupling or there is Raman coupling is relative weak I see that so normally what they would do if for that no coupling I would have just three cosine that are shift one another by the the flux but now if I couple them I open up a gap a gap between them but if I'm far away from from the crossing this awaited crossing the the color the raspers and the spin remain more or less untouched so it means that you see we can achieve a situation in which we have a gap and we have state that leads in this gap have almost linear dispersion relation and and well defined spin so these are good through a prototype of a edge state indeed we test that and here that these are a good transmission property but this and we test also in the experiment so one of first experiment the realization was at least with young Spillman group with rubidium and then they observed the the skipping orbit and the carality so that one one spin state on one species going on one side we see from the dispersion relation and the other is propagating in the opposite direction and also the other experiment this was at length with fermion and also Marcello and Peter solar were involved in theory and here the system is slightly different in a sense that they use different spin state this atom as we learned yesterday as six internal state but again the the similar carter behavior of the side state was clear of self okay then the win very recently also this has been settled to the clock state so this is a similar behavior can be as we will see also seen in the ladder and this has been an internal lens and a gila okay fine so maybe yes or maybe not I convinced you that even in a very narrow system we is listed to to speak about that state in the sense that this have the good property to be associated to a state in this week limit but what about the bike so this I it's a state that has just three internal state so our bike would be just one side so it's this minimum for one it's even hard to understand our attack this problem because it's really the general question with how big should be a strip to see some topological feature and if we are we have H state then we said that is this open close boundary correspondence so that the property of the state should be related to the property of a bike of the band to the turn number so is the turn number somehow defined can we measure it in which sense so the practical way of doing it it's so what we consider in this paper that appear in August in Cyprus is published in August in Cyprus this new new journal that is scientist for scientists so it's we took a pragmatic approach and so what we measure we wanted to see if in this system we can establish a laughing pump so we can apply in the force if we can see a displacement of charge that is related to the to the turn number and in this can work also in the small system so this was it's related also to the tallest pump that was discussed by Michael and so consider the now step behind so consider a more typical situation a safe situation in which we have a large periodic system which we can have a well-defined villain zone and we can suppose we can prepare in the lowest band a state that is well localized in in the transversal direction that they call why and and well spread in the in the standard direction that is in the large direction that is sex in this case even is not there is no distinction between large or small but we'll be afterwards and now so we have this state and now suppose that we apply a force along x so semi-classically we expect that this way packet with accelerate to start spanning our brilliance on so we wait a block oscillation this state will cover the full brilliance on and will and will come back so what happened in this case is that if there is some topology here that will be in a in a normal velocity that was the center of mass imposition of this state okay so this is the the sketch how this would be in formula so again so we would have this is applying the family the famous tallest argument so we will have the semi-classic question of motion we will have the momentum that is growing linearly with the the force and now as look I say the group velocity depends on the dispersion relation but when we are in a topological system there is also an anomalous contribution that is due to a force this is just really Lawrence force in the brilliant zone this would be the interpretation of this curvature and if we have a way packet of the kind I said before so very very localized in in momentum along x essentially this doing the pump along a period so the time in which we span the the the brilliance on in X is exactly equal to integrate the expectation value of this velocity averaging it on the brilliant zone and this means that this guy disappears if the band would be full and in the end the only contribution we have is we have the integral of the curvature that by definition is the share number so means that if we if we just measure the center of mass displacement what we observe is that this will be transversal as we expect this quantum also the current is transversal to the force and will be exactly of the quantity that is proportional to the certain number and now the point is that this kind of state I present you that has this shape it's it's very easy to prepare if we are in in a limit in which the the coupling along wise it's so this data are well decoupled so we can prepare a well easily a well-defined speed so in the limiting which this is zero would be just having in this synthetic idea just having a condensate that is not couple and also what we're showing a means that this indeed it's applicable also to a strip so some a system that at boundary until we don't touch the band so the formation that is produced by a boundary is only seen that you are really close to the band so let's see how this apply to our system so let's take again this three three internal state so ideally if we start without coupling we have this situation in which we have just these three band that are displaced and preparing a well-defined so localized the system and in why means that we localized it in spin and means that we put it at the minimum of this personal relation now consider that we turn on just a bit the the Raman coupling so the situation become this you see we have this weather crossing the opening here but as the color as seen from the color apart the the crossing if the coupling is weak the color remain well defined so in the minimum more less we are still the condensate is still occupying the as a well-defined spin so it's still occupying a well-defined site in the synthetic dimension now what is doing this this pumping we were saying so this is really in a in a period in the period we consider it just the movement from one minima to another so if we move from one minimum to another in this approximation in which the the center of mass it's really the average spin is really well-defined this is really an integer so essentially the interpretation here is that the same number is just this the the number of spin you have to jump when you move from one minima to another indeed if you go back in the article of the feminine the seminal article of tollers this they consider exactly this also this very weak limit and observe that indeed the chair number can be related to the solution of the deophantine equations is sound complicated but the deophantine equation is exactly what you ask if you want essentially the minimum here is that the cosine if the argument of cosine is a multiple of two pi so you want to know if the as this this minimum cosine will be just the momentum shift by the spin for for for the flux essentially this is really related to to the value of the flux so let's consider in this case what I consider specifically is a flux that is a rational flux because as I said that the maybe I didn't said but the to have a nostril spectrum we need a rational flux so this is the simplest one of the simplest we can imagine so just one third of the total unit and in this case we see that with the relatively weaker coupling in the way direction the pumping is working very well this is would be the trajectory of the center of mass in x they are not so interested about but what is really relevant is the center of mass in y so while we do the the pumping after a cycle we move to one then when we are here what's happened essentially now we see do we have this this one that was the edge state so this is periodic so this is pumping up essentially to the second band and now we start pumping in the second band and we see that in the second band and this is well known that in the lowest band the turn number is one so we displayed by one but in the second band is minus two for these values of the flux so essentially we go back and then when we read again when we cover all this band we are pumped up to to the higher band and again this is turn number one and when we cover after nine step we cover all the all the all the spectrum all the bands and we are back on on the on the original sites okay so what is quite impressing is that the prediction one is getting doing evolution is almost really for an ideal system that is infinity in the x direction is almost one it's 90 99 or 90 or a point and then point 98 okay so the natural question is why this is working so well so again the crucial point is that this this gap as I said it's it's hoping linearly but the mixing in in the spin state is quadratic and indeed what we verify is that the degradation of our measurement it just depends if we assume that we are able to be always sufficiently adiabatic so this means that this means that one should be very slow compared with this gap otherwise we are we are we are going to the higher band and we don't follow the lower band so if we are in this regime really the the measurement is just affected by the spread of the center of mass and this is going quadratic essentially because this is just a simple perturbation theory okay so this is for for the simplest case you may say okay this is a sequel one but in principle the same argument would work for any for also an higher trend number and if we are in the lower band what we ask is that the number of what we ask is that we don't want to be pumped from one edge state to another estate so we ask essentially to have two additional two additional sites so we test also for other flux and bigger system and this is more for the experts so curiously even the this pumping is working back the the Fukui Atsuugai Suzuki algorithm that is a very efficient algorithm for computing numerically the churn number in the Brillouin zone and for example for example for this value of the flux that give churn number minus two the algorithm and the pumping give the same prediction until a number of spin state or the transversal state of five but when we go to four we can still perform perform the pump so we can choose a well-defined minimum which we can move in the lowest band and observe the displacement of minus two while the algorithm big down so what we test also in this paper is that this system as should be it's a robust to disorder what does it mean obviously robust to disorder that have a magnitude that is less of this gap that is tiny this is important and also we put also an harmonic confinement this what is giving you is giving you a lower bound on the force that you can apply this means essentially that also is telling you that you cannot be as as small as you want in in the coupling in the in the synthetic dimension but still we we see that we can still get there is room for experiment to get a good number and work in the in the right regime now what happened with interaction so one neighbor say the gap is small so the this should break down but indeed the one can we'll see later that there are adiabatic argument that connect this the physics in the strips and this has been studied also by people here by Marcello to and Rosario for for for considering them how connecting to laughing like state so also in this case and this pre perhaps it is more to study in this this contest we can still find find a a precise connection between the topology of the small system in the large system also with interaction and in addition again here from from a chest and we show that this kind of a narrow of the steep but also a symmetry protected 1d topology that is not the one consider here in a sense this is the topology here is in any terms of the the one of the two big system but it's interesting that this kind of system have also a bounce date that appear now at the extreme and this appear in on August okay so how much time I left okay so we'll be I'll try to give you a sketch so what happened now it's very interesting what happened when we have we have interaction okay this is this synthetic lattice seems a promise promising route not eating in principle is expected there is a peculiarity that is quite obvious that the interaction in this in the dimensional range this is gives quite nice new feature and contractually there's also not only a practical but also theoretical tool to connect what happened in a sense why why when a 1d system become 2d so this is and in a lot of study and also a lot of your interest and here what to present you the effect of the demonization so let me tell you that already one is going on a ladder that is quite interesting physics so was discovered or point out a long ago that even just this two level system as a nice analogy with type 2 superconductor why because okay you have you have a flux and you can have depending on how is this transfer coupling and now I call J perpendicular you have two situations so if this coupling is sufficiently strong you see that in the you just compute the diagonalize the 2 by 2 amiltonian you see that there is a minimum in the center and this is essentially the analogous of of a Meisner phase so the only thing that happened that you have circulation on on the edge and the and the magnetic flux essentially is not able to to pierce the is repelled by the the superconductor now if you increase the flux keeping this transfer coupling you see that a certain point that the minimum split and the the situation that has to minimize associated to the fact that current inside appear and this current slow let pass some so in the analogies that in the type 2 what happened is that if the flux is 2i vortices that appear in the system and let pass part of the of the of the of the magnetic field so there are part of the system that become become a remain superconducting but other part that are a normal conductor and this is signal by the presence of the current around along the the rank of the of this this ladder and this is also been observed in the experiment in real ladder experiment in the in block group okay so what happened is one is putting interaction very briefly normal interaction would try to stabilize the Meisner phase so reduce this effort of the Meisner and also was shown that in the synthetic latter lattice this is even more extreme because somehow we have this interaction that is more because we have also interaction on the rank but still there are more phase visible if we go out of the now what we we wanted to to see that there is another way to favor this this vortex and one possible thing is demerizing the lattice so what does it mean demerizing the lattice so suppose that you instead of having just a normal optical lattice use you have you do a big chromatic one in which essentially create a situation which you have a strong link and can and the weak link so that one one the tunneling desire and the other is weaker so you see that the fact that you disconnect if you put this coupling to zero do this connect in placets and this already tell you that is like having isolated the Meisner region separated by by a vortex region and when you turn on the coupling you would expect that somehow this this vortex if you have interaction should not ok but this the next thing is that the the physics is interesting also without interaction because ok this first of all how it looks like this was the original system now I demerize so essentially I should fold the madman structure this I just changed the way in which I picturing it in the restricted building zone but if I demerize it ok this I just rescale if I demerize it what is doing the demerization opening this gap here but now it's also separating these two minima so you we see that this naive expectation is confirmed so that this vortex phase is enhanced because we were the minima are more separated so we we get more fat from the Meisner phase and this can be quantified within really looking at the reverse of the chiral current so the Meisner phase as you expect what does it mean means that if you are in Meisner phase you're applying a flux that is increasing you expect that the current on the border is increasing on the boundary of the system so along the legs so the chiral current is just the difference between the the current that is going on one edge and the current that is going on the other edge so they are opposite sign because they are chiral so they are that but so this is increasing and then in reaching the maximum when the minima starts to split and we enter in the vortex phase and they start decreasing and this is the behavior in the uniform ladder now what happened if this is easy and tricky analytical result but can be done everything analytical so if we put the the the coupling we see that again we enter in the this vortex phase a certain point and also we observe that more we demerize more the water phase shrink but then we see that there is also this this unexpected a priori behavior in which the the current is not only decreasing but is changing sign so it's going in the other way around that is increasing again so we have a new the magnetic region in which this is positive the magnetic region would be here even and then that where there is vortex so this is a bit more really like a superconductor and then it's decreasing again and so this confirmed also the with the current behavior confirmed that this vortex phase is enhanced so just flashing you what happened when we have interaction so we we did essentially both analytical and numerical studies so the the obvious analysis static you can do you can start when you disconnect really lattice and you have just this connect placket and you do perturbation theory so you start looking at the spectrum you can derive from it if you do perturbation theory and effect xx model that but already from the spectrum you understand that if the coupling is too big you have just a bending cylinder because you are in the in the in the situation in which you have just two atom per placket and they cannot overcome one another so the system is not conducting but if you decrease the the coupling you get in a phase in which you have just one atom in this placket that is circulating with the flux and this is the the in that you you have what the phase actually there the vortex appear in the in the weak placket if you like and this indeed what is observed also with the energy that if we start in the situation in which we thank you and to have at zero flux just this to state the generate we see that we enter at zero flux we enter immediately in the in this vortex region so the both the the energy and the density decrease if we have more flux this is getting a bit more because this vortex it's more energetic and the the the behavior close here can be indeed it's explained by by the the perturbation theory and the main result would be that also not all it is but we can show that by the realization we can see clearly sign of the incommensurate incommensurate phase transition and this can be signed for this in an observable quantity like the the density the one the one body density matrix or the the density of state the number of particles momentum average on the two legs and the transition is just this change in behavior from these moods Lorentz and distribution to a peak one this has been observed also sorry this has been observed also by other people for example or in yacca and Roberto Cittro and here in Trieste the final Apollo but with finite interaction so here this feature seems more robust and and also it's quite robust to trap so as I have no time there are a future step obviously going out of the our core our core boson and see if the this demerized lattice can be can be used also to make more visible sign of a laughing like state so this was just a summary of of what I tell you so this water they all work we have we can add a synthetic a state that being a server in an experiment but also bulk topology can be observed and then if we go in with interaction the effort of demerization could be very nice tool to be studied and this is some possible direction but let me just conclude thanking the the collaborator so this start to in Barcelona when I was still at a to be with a casino sila torre what have you bought that was the student was supervised and much a cleaver stand and I moved to week from to be as we work with what happened when we put interaction in the but without flux then with these people we studied the the case of the synthetic how you get a state in this synthetic lattices and then we study also non-trivial topology of the mob you strip with I didn't say about it then more and more and very recently okay also let this year Alexander joined for understanding this this this topological property and in the last work that is as to appear manuel is the driving force is the guy that is doing the energy and a robert is we learn from a bit more about this interacting ladder and also non-analytical method in this system and the other is still pace so if you want to join me you're welcome thanks a lot