 their cavity point for dynamics. It's a little bit different from the official. Yes, it's my fault. So thank you. Thank you, Giuseppe. Also thank you, Adriano and Pierre, for organizing this. I really find it very useful and informative. So thanks for taking all the pain. So I'll just start with my presentation. This is the work done in collaboration with Rebecca Kraus, Simone Jaeger, Giovanna Morici from Cybrookin and also with the Marseille Rochelle Day from Lyon. So the idea of this problem comes from this experimental setup. And what happens here is you have a two-dimensional optical lattice, a Bose-Habban model, and you insert it between two mirrors, which is a cavity setup. And if you pump this setup transversely with a laser, the photons from the laser can bounce of the atom into one of the cavity modes, which gives you another wavelength in the system. So you already have a wavelength lambda naught because of the optical lattice that you have, the Bose-Habban model. And on top of that, because of this cavity with the standing wave, you have another mode which gives you a wavelength lambda. And the competition between this lambda and lambda naught, as I'll show in the next slides, will give you new phases in the system. So what happens in this whole scenario of having your optical lattice and then another lattice on top of it because of the cavity interaction, in a specific limit of time called adiabatic elimination of the cavity, you can write your long-range Hamiltonian in terms of three different parts. The first part is the Bose-Habban Hamiltonian, which will stay even without this cavity setup. The second part is the laser term, which is just a potential shift because of the transverse laser that you apply. And the third part is H-cavity, which is the long-range potential mediated by the cavity photons interacting with the atom. So atom photon interaction leads to this long-range part, H-cavity here. Now, the setup looks much more clear here in this figure on the left-hand side of most coordinate. And different colors I will now use to represent different parts of the Hamiltonian. So HPH, I show in red, which are these red dots with wavelength lambda naught. Gray H-laser would be the Hamiltonian of the laser and blue would be the cavity, the cavity part. So the red Hamiltonian, HPH is the Bose-Habban Hamiltonian, which most of you already know. I'll just go through it still. The first part here shows the hopping when atoms can hop from one side to the other. I now show it with the strength J, but actually this, I mean, I write it as the strength T. So just consider they can hop from one side to the other with strength T, with tunneling T. When they lie on top of each other, they pay an energy cost given by U. And Mu is the chemical potential. So it takes care of the number of atoms in the system. H-laser is this gray part here, which is this transverse field that you're applying. And this is just a potential term, which is just this modulation given by cos square kx. Wi's are the vanuir functions for the site I. And H-cavity, the blue part, you can also see that this is a modulation. But then because of this n-term here, this is a modulation in the number of atoms that you have. So it's kind of a density modulation that you have in the system because of this cavity. So you'll have a profile of how your atoms will arrange because of this cavity-photon interaction. And together this Hamiltonian take this form, H-l-r-i. Now these cos kz and cos kx, as I told you, will depend on this lambda, lambda naught and lambda. And what happens is these terms take a form as a ratio of lambda naught by lambda. So you can have two different scenarios. One, where the ratio is an integer and you will see that in cases where, for example, this green line where it is an odd integer, you will see that the potential goes from plus 1 to minus 1 on alternate sites so that it's more like a checkerboard pattern because of this potential. Whereas in the case where the ratio of the two wavelengths is slightly shifted from an integer number, you'll have a quasi-periodic potential and the profile or the envelope of the potential is given by this red curve here. Also, I'm reminding you again, if you have any questions, just stop me and ask even if I cannot see the question on the session, just like anyone from the from the panelists stop me and ask questions. So as speaking of Shradha, I have a very short question. Yes. If that's okay. Sure. Which of the two terms gives the red envelope? Okay, so it's actually, so when you take this, yeah, so when you take this to be like slightly shifted from this, you substitute it here and then solve it and then you will see that the red envelope is this cos 2 pi ix by ip where ix ip depends on now the part calculated because of this ratio. So you see the red envelope is this term that it appears here. I see. I see. So ix are actually the site index. So you will see that as site changes, it has like, I mean, an envelope of checkerboard pattern. So you have like, so you have potential going from plus one to minus one, but in this profile of this modulation. I see. Thank you. Okay. So I mean, these two are an interesting topic on its own, but I will just concentrate on the common straight lattice part where because of this long range potential being being so because of this ratio being an odd integer, we will have a checkerboard pattern which leads to phases like charge density wave and super solid on and also we have phases like more insulated and super fluid, which would have been there even without the the laser part. So, so I mean, this is these are the phases that you obtained in the both subat Hamiltonian. And because of this, this modulated modulated long range potential, you also have charge density wave and super solid phases. The Hamiltonian now takes a much cleaner form because the laser part just provides a potential which you can add up here in the in the cavity term and doesn't really affect much. So now you have a both subat Hamiltonian and this cavity part, which because of the ratio being being an odd integer takes the form of minus one to the power i. Experimentally, you can obtain this parameter here by looking at the cavity output photon number. And therefore, this long range part of the Hamiltonian is shown in green here, which is just the difference between the occupation of the even and odd sites. Now, just to give you a visualization of how these phases arise, the the red term the red underlined term here is just the tunneling part, which gives you the super fluid phase, if it if it if it overcomes all the other terms of the Hamiltonian. The blue part here is is the one that provides you just the mod insulator phase, where all the particles are stuck to the minima or maxima of your potential, we'll consider it to be minima and then you have the the the the the long range part with the with the U naught term of the Hamiltonian, you can get a charge density wave, where you have a 1 0 1 0 or like or 2 0 or occupations like this where where you have an imbalance between the the occupation in even and odd sites. There can be another phase where the long range term and the hopping plays a part, which we will refer to as a super solid, where you have a long range coherence in the system, but it's also modulated and there is an imbalance between an even odd even odd occupation. We will look at the mean field analysis of the system, and then we will also compare the mean field results with the with the quantum Monte Carlo, but just to begin with it's since it's a long range system mean field should work very well. So, we will also verify that to as you can see that in this Hamiltonian the n i terms are can actually be written in the occupation number basis as the diagonal terms, but because of this long range interaction and because of this hopping, you have a much much higher terms than just the quadratic part in your system. So, we will decouple these two terms, the red term here would give you a super fluid order parameter, which is just the just the average occupation of the annihilation operator and the green part here would give you an order parameter, which we'll refer to as the imbalance order parameter that this tells you about the the the cavity effect being present in the system or not. And therefore, keeping with or applying this mean field, you have a Hamiltonian, which is now side dependent and you will see that that it just has two terms in it for the even sites, you can solve it and for the odd sites, you can solve it separately, so as to get get your result. What happens to these two parameters in different phases is in mart insulator, you do not have cavity effect and you do not have the super fluid order parameter. So, these two parameters should be zero. In super fluid phase, you have long range coherence in the system and therefore, this phi i term is non zero, but then it can exist even without the long range attraction. So, theta is zero. In charge density wave, you have phi zero because you have a modulated fixed insulator pattern, but your theta is non zero because this is the term because of which you have this modulation and in super fluid phase, both of them are non zero. So, this is how you identify different phases. We choose two different values of the long range interaction 0.3 and 0.6, 0.3 being the left panel and you will see that for you are being 0.3. So, the topmost figure is the plot of the super fluid order parameter phi and you will see that the super fluid order parameter is non zero very clearly in the insulating phases like charge density wave and mod insulator, but it's also non zero in a very small peak here which is which we would refer to as the super solid phase. Theta is the cavity output which is non zero in the charge density wave phase and also in the super solid phase. So, this is how we know by looking at these two that this is the super solid phase and the same for the U L is equal to 0.6. If you need more details about the mean field and this work it is in this paper by Lukas Himbad. Now, to verify the validity of the mean field results, we compare our results with the quantum Monte Carlo results by Flota-Tetal and we see whenever there are long range phases, the phases because of the of the long range interaction, our results match qualitatively very well with the with the quantum Monte Carlo results. Some transition boundaries may change at higher densities but that is also because in that case you also have the phases because of the short range interaction which are super solid and mod insulated which does not even have the effect of the long range part and therefore mean field is not really the perfect or like the best way to compare with quantum Monte Carlo results in that case. But in this case where you have mainly the phases that we are trying to compare charge density wave and the super solid or super fluid super fluid phase, we can see that our boundaries matches quite well with theirs. We also find a phase separation region PS here. This is the region where for this special density you do not find any solution. For them it's like is the sampling problem that they do not really identify these two phases particularly. So once we have matched our mean field results, we ask the question what we can do extra to really look at the behavior of entanglement entropy in these phases and mainly to ask about their scaling or what we can say more about their their excitation spectrum or kind of behavior that these these different phases would have. Because mean field theory is actually just like for us defined for a single site, we add higher order of correlations in it by just like by doing this small procedure named slave was on approach, which is mainly taking from this PRL paper here. But there are like many papers already, already using this approach. So the idea is you have the solution from the mean field Hamiltonian for each side, which would act as a new basis where psi alpha where alpha goes from zero to n max n max being the the number of the occupation of the local Hilbert space. And using this psi alpha you write your original Hamiltonian rotated in this mean field basis now. Now what you see that there are higher order terms in this gamma operator here, which is the the operator obtained by by having the by having to write the right the the the occupation number basis in the in the mean field basis. And you just expand about these dramas and truncate up to the quadratic orders you expand about the mean field ground state and truncate up to the quadratic orders so as to obtain a quadratic Hamiltonian. Once this happens, you can have a quadratic Hamiltonian which is much cleaner to look at. And also the correlation would consist of only one body correlation ci dagger cj and ci cj in terms of which you can write your correlation matrix for the specific subsystem that you're looking at whose eigenvalues would give you the entanglement entropy. With this information we look at the the the entanglement entropy for whole for for whole range of mu versus t both rescaled by short range interaction u0 for two long range interaction terms ul being 0.3 and 0.6. We also consist a torus geometry where usually we take this subsystem a to be half of this this torus but this is just to show that we can just like change it. With this in mind if I look at the the results for ul is equal to 0.3 that I see that we see that for a fixed density if one starts inside a charge density wave phase or an insulating phase and go to to to a super to a long range interacting a long range coherent phase. So for example for rho is equal to 0.5 we start in charge density wave and end at super fluid phase and there is no quenching here we are just changing z and looking at the the the value of entanglement entropy. We see that entanglement entropy would always show peak just at at the transition from incompressible phase to a compressible phase which is which is also there for the for the mart insulator and was like first shown for the mart insulated case in this paper by by Tre-Roth and Rochelle Day. We also see the same effect for for different densities so being fractional density or the or or or an integer density. The same is obtained for ul is equal to 0.6 it is much easier to see it for rho is equal to 1 rather than rho is equal to 0.5 because there is a long phase separation phase in between so we cannot say much about this phase. So so we always see that entanglement entropy does not really tell you much about the transition from the from the super solid to super fluid phase but it would always show a peak at the transition from an incompressible phase to a compressible phase and the and also we see that this this peak happens at the fixed density but if I define two points say for example b is equal to 0.5 being at the fixed density and e is equal to 0.5 away from this this fixed density point we see that the the the peak goes away because this lies mainly in the in the in the in the excitation spectrum of of your of your system at these two points. So for b is equal to 0.5 you can look at the the ground and the excited state going linearly sorry linearly to 0 close to k x k y is equal to 0. So these are the plots of the excitation spectrum for for k x 0 plotted with respect to k y at this point b 0.5 and you see that both the ground and the first excited state goes linearly to 0 whereas for e is equal to 0.5 slightly away from the fixed density the ground state goes goes not linearly but rather quadratically to 0 and there is a gap between the ground and excited state. So this when you try to extract the the entanglement spectrum from your from your from your correlation matrix you will see that the spectrum also has a different behavior and actually kind of same as compared to the to the physical spectrum and this tells us the the the reason that the constant density phase transition at tips are actually the special point because you have a different critical exponent z being one at this point and therefore these two points belong to different universality class and we have a different behavior of entanglement entropy for these two points here. Also the sorry Shradha but so as far as I understood from the dispersion relation between the ground state energy and the first excited state energy at the tip you have a second order phase transition point and so critical point whereas all the the rest of the phase transition seems to be first order no no so actually so e is equal to 0.5 is the second order phase transition it also the first order phase transition you're right exist in these systems but that would be actually like very close to this boundary here so actually this is a very good point so in this system it's really like very weird that if you go directly from this charge density wave to super solid super fluid phase here rather than so for example here you can see very clearly so if you go from charge density wave to super fluid to super solid there this would be a second order phase transition but if you directly go from charge density wave to super fluid phase here you will see a first order phase transition okay I mean so this point actually is kind of here and this is a second order phase transition point but but then how come that there is no gap closing in e 0.5 so see I mean this is because I'm taking a very very shitty system size I mean there is so you can also say it from here that it's not very clearly again gap closing here also there is always a finite gap but then that is because I'm taking like 20 cross 20 or 10 cross 10 system and actually these can be done for two cross two very well because you know the main field is just an even I mean we do a main field for just even outside I can just do it for a two cross two system but then you know you will not have this behavior with k x k y so it's but but so what you're saying is that it's just a numerical artifact that I'm not doing it like very well okay so what you're saying if I understood well is that if you were to take a bigger system and really go on the phase transition line for e 0.5 you will have a gap closing at k equal minus pi over two and pi over two yes yes oh no so here I don't know but definitely at k x k y is equal to zero why do you say that it should be at minus pi over two and pi over two the the blue curve is not the ground state energy is the first excited state energy no blue is the ground state energy and red is the first excited state energy but why should a gap close here but a gap is the energy between the ground state energy and the first excited state energy okay you're saying because it is at the transition line so at the at the end of the so let me just think if I open this blue line zone for example for the for the for the mark insulator and the super fluid transition you would see the same behavior away from the tip for example at point d1 and here I've just like like because of the even order symmetry folded the blue line zone now I have to think why should this gap go to zero at this this sorry maybe maybe not at this point maybe k x k y is equal to zero but the gap between so this is the point at this point the so if you if I would tell you the critical exponent that would be how the how the gap opens and here the gap if I'm if I'm understanding correctly would be the gap between this e and minus e here so as away from this point you will see the gap opens as z as k to the power two so z is two and in this case there is also a minus e here and so it's more like you see that but so I think it's more like this that the that the goldstone mode and the Higgs mode both go to zero at this point and here this does not happen okay but the gap is actually the blue curve is actually the the the lowest unoccupied band yes okay okay okay sorry I'm sorry I wasn't very clear when I said the ground and the first excited state here yes okay um sorry I also have a question if I can and this model that you are considering at finite interaction um as far as I know as an exact solution when you take a strong repulsion limit because it's reduced to an xx chain with anti-ferromagnetic infinite order chain you know interaction so you're saying if I take uh ul to be very large because the both of them are here right so I always see that when I take ul to be very large this is unspoken no ul u zero the the repulsion the on-site repulsion yes that basically becomes an xx chain plus an infinite long range anti-ferromagnetic coupling which is exactly solvable reducing the the space of modes into two-sector yeah this can be done yeah and I was uh I was wondering whether you compared your funding with uh in the live I mean if you reproduce the correctly the limit of strong repulsive interactions actually this is a very good point and I should look at it but I did not check it we actually just checked I mean I just okay if I put ul is equal to zero I I match all the results with the Bose urban model but I never checked this this would be actually yeah there is a paper of igloy I don't remember the year it's on here be probably 2018 something like that I will write your mail afterwards too yeah I know and also in that paper coming back to the question of Pierre there is also they made a number the Stratovich transformation so that they could they were able to characterize the face diagram of the model so there you can find all the transition line okay okay maybe it's not the same at finite interaction but I guess so it's just a bit deformed I guess but yes I mean I would believe so yes okay thank you but this is a good point thank you I will get more information on this okay I mean just to say that these two points I mean this is true that I haven't found this point e explicitly by by solving this equation I just numerically found this point where the the the curve was changing the excitation spectrum was changing deep in charge density wave and in superfluid phase but you can actually find these blue lines analytically by doing a perturbation over t and these blue longitudinal lines between different phases can be found very exactly by because you know you start from here by putting t is equal to zero and then you extend these lines for a for a fixed density transition so I will check this I will do it much better but like this being non-zero is just a numerical artifact with this in mind there is also another interesting thing happening here because of the the the long range or the infinite range interaction here the k is equal to zero mode plays a very important role here and you will see that that for for points in the super superfluid phase c 0.5 or in the super sorry in the super solid phase c 0.5 or in the superfluid phase d 0.5 you will see that if I plot for kx is equal to zero k y ranging from minus pi by 2 to pi by 2 you will always see that there is a mode softening for k for the for the for the higher energy at k y is equal to zero so for kx k y is equal to zero there will be a mode softening and this is solely because of the infinite range interaction here and if I compare the the the energy slot here or the energy spectrum here you will always find a root on mode at the super solid superfluid transition at the at the transition of of this long range long range phase and this has been seen experimentally also but like sadly this one mode and this is a good point here you know this point should also actually go to zero I asked to the for the other group who does it but they do it like explicitly for a 2 cross 2 system to look at how the ground in the excited state go so I mean our main point is to see how the entanglement entropy or if at all this would show a slight peak for just mode softening of this one mode and at this point I cannot comment much on this but if I look at the entanglement spectrum and how different modes contribute I see that the entanglement spectrum wouldn't change much when you go from a super solid phase to a super fluid phase so in my opinion this does not really show a peak for a finite size system maybe when you go to a thermodynamic limit explicitly you might see see a special point here where you can see maybe a non analytic behavior between I mean a peak between a super solid and super fluid phase just to say comment more about the physical spectrum there are more interesting things happening when UL is greater than 0.5 so we take UL is equal to 0.6 and the main idea of doing it is because because you can see from this this figure here that the super solid phases are so large that you also see a transition between different super solid phases here and so in the in the top left left corner we plot the excitation spectrum so this red and blue curve is actually the excitation spectrum for kx is equal to 0 and ky is equal to pi and we see strange points a and b here and if we actually now trace back it to the phase diagram these a and b points actually lie at the transition from so a point lies at the transition from one kind of super solid to the other kind of super solid and b points lies close to a triple point and we can so I mean this I have been found experimentally someone saying that then we can say that if you really want to probe a transition between two kinds of super solid you can actually look at the the the the gap between the higher excitations going to 0 at ky is equal to pi by 2 so I mean this is just to say that that you can also probe different kind of critical points for example a triple point by looking at the gap going to 0 at pi by 2 and you can also obtain a data cone kind of like structure here. We also I will very shortly comment on the metastable states and include like just the just not the static effect but a quenchic kind of effect and the idea is these systems are like are so complicated that there are many metastable states in the in the compressible phases incompressible phases here and if you quench from so for example even within the the more insulator phase if I quench my parameters I will see that I might end up in a in a in another charge density phase and here we compare our results with that of the the results from this experiment where this whole idea started from so in this case they actually so here they very clearly show the phase diagram between the detuning at the depth of the lattice and the detuning is related to the long range so detuning and depth both are actually related to the long range interaction terms and the depth would also tell you the about the the interaction part because you can increase or reduce the depth to see the hopping and reduce the short range interaction so the so the idea is they have the whole phase diagram fixed to to the to this density is equal to 1 and they see that that this slanted region here is the is the so in their experiment they see that this is kind of of an overlap or a coexistence of a charge density wave and a multinsulator phase because whenever they start from somewhere close to the multinsulator phase and quench inside the charge density wave phase they see that they need I mean this makes sense that they need more potential to really overcome these two incompressible phases so that at the end when they come back or they try to come back to the multinsulator phase they always see kind of a hysteresis behavior so so I just explained these so this s is the super solid phase this is the charge density wave phase multinsulator and super fluid this is the coexistence region and in the plot just below it they show behavior where the star is the starting point they start from some point of the detunic and then they they change their their field reduce it and increase it so that they they for example start from here coincide and come back and then they always see a hysteresis behavior which is because of this potential that I said that you need to increase uh much larger than some potential so that you can you can maybe come back to your original state and if we plot the if we plot the metastable states in between the our our phase diagram uh comparing again u l is equal to 0.3 and 0.6 and keeping fixed to this n is equal to one region or the occupation of n is equal to one we also see that even if I start from say some point in this loop in multinsulator ground state phase diagram and I quench my long range interaction to u l is equal to 0.6 to this point and if I come back I can end up in some some uh some charge density wave phases that we found find inside this multinsulator uh multinsulator mode and this paper is really interesting there are many many more states that you can find which are these like uh this orange and red region where you can also get trapped so in that case we match our results very well with the with the experimental finding also with this uh with this mean field and doing an expansion on top of this mean field solution uh we also look at the scaling of entanglement entropy at different phases just to see how entanglement entropy would behave mainly in this long range phase the the super solid phase and we again take u l is equal to 0.3 fixed density 0.5 uh this is the form of the scaling that we that we take for the entanglement entropy uh the first part is actually the the uh area law term the second part is the volume law uh sorry the second part is the log correction the third part is volume law in case we obtain it and then some constant and we see in the charge density wave in both the cases uh there is just the the this constant so we take d is equal to 2 it's always a two-dimensional system there is always this area law term and uh no log contribution or or obviously volume law as expected in the super fluid phase also this happens that there is a log correction on top of the area law but then we do not find any volume law which which I don't know was expected or not so it was good to see how this would behave uh we see the same and in the super fluid phase again we obtain an area law and a log correction so the log scaling was was kind of expected for the super fluid and super solid phase but we did not know if there was a volume law scaling at at this like static level or just at the level of this long-range interaction of the super fluid phase so to conclude uh we have used the slave boson main field approach to probe entanglement entropy of a long-range porcelain model with phases like charge density wave super solid mod insulator and super fluid entanglement entropy shows a peak at the fixed density incompressible to compressible phase transition but this fixed density may or may not be an integer it actually depends on the on the excitation spectrum and there are these special points which belongs to a special universality class which keeps rise to this this peak in entanglement entropy fixed density in incompressible phases have an additional log dependent scaling and we observe area law scaling for area law scaling with a log correction in the super solid phase we also have some results for the incommensurate lattice case if we have time and you guys are interested i can i can talk about it but then it's also interesting to look at this this incommensurate lattice case uh these are the people who worked on this lucas is the his master thesis was was to look at this mean field phase diagram and we actually did this meta stable stability work with him uh simon rebecca and juana and tomaso are the other authors in this paper with this i thank you and questions thank you very much shahda for this nice talk are there questions comments sorry if i can ask something in the slide where you showed that your model has a super solid super solid transition what is the difference between the two i suppose the crystalline structure sorry which one i just wanted to be sure this one uh to super no the super solid to super solid one that you mentioned at some point i think it's a slide before oh yes yes it's because of the occupation so you see this is the super solid of one zero so you will have a kind of a so how do i explain it so if you look at this density profile you know you can okay i think i see so it it's the same crystalline structure sorry no go ahead no no so you're actually absolutely right is the same crystalline structure as one zero or two zero but then there is a higher um uh or the the higher probability of of even side being occupied with respect to the outside so for example here we can have an occupation like zero point eight uh zero point one point two and here you can have an occupation like uh one point eight and point two something like this so you will have a different modulation because this is on top of this charge density wave which is two zero so you will see a slight modulation from this two zero here in this phase uh also you can differentiate this phase very well by looking at this value of theta yes so the crystalline structure is the same it's it's just that the occupation of the sites in the structure are different yes yes essentially exactly okay so in this case even side is again preferred than the odd side and the the odd side is actually more close to zero so the crystalline structure yes is same the occupation differs okay thank you so if people don't ask questions i'll start presenting other results for the incommensurate lattice case so i i would like to to to ask you a question so just the point that i that i'm sure you mentioned but i might have misunderstood so when you move mu here like for example in in some of these um in some of these graphs do you also move the feeling or is it just the amplitude of the chemical potential yes if i move so for example for a fixed t if i move mu here the feeling would change okay but so it means that the the feeling seems to change continuously no no that you have continuously sorry no uh why do you say it would change continuously because from going from one zero to two zero this would be a first order phase transition and filling would also change uh discontinuously right i mean you suddenly rather than filling one because we have a gap phase of course yes because we have a gap phase so so you either you have a full band that is empty and then suddenly fully full but you don't you don't consider the in-between case yeah okay okay no no good this was the correct way to put it yes because you're and actually the transition between one kind of super solid to the other would also be a first order phase transition especially close to the boundary of this charged density wave to super solid transition mainly here it becomes second order close to the super fluid phase okay close to these kind of so now if you were to change not the chemical potential continuously but the feeling continuously then you will have some of these commensurate results and all of the incommensurate results in between i guess uh no so in this case you will not have incommensurate results to have incommensurate results you will have to have this kind of potential and therefore the Hamiltonian would have all the terms here actually so to have the incommens I mean in this case or with this Hamiltonian you will not have incommensurate results now because you can always since uh since u1 is uh is conserved in your model you can always put the amount of particle that you want that can be incommensurate in your system and then you will not have a commensurate phase but maybe this is not the incommensurate incommensurate case that uh that you studied maybe it is another case I mean another case of incommensurate next actually you will have to repeat it again I am sorry I no no no it's uh I mean if you were to put uh an incommensurate number of particle in in your in in this model okay so with uh with a well okay I understand you're right so for example close to this boundary uh more insulator boundary you'll have some density like 0.9 0.98 so I mean that you can say that there are incommensurate okay you're right yeah because the sf is uh is uh gapless and so yeah yeah you're right you're right so the the feeling also very continuously as you vary the chemical potential yes okay everything is very clear okay thank you very much I confused you in the first part you're you're right there would be incommensurate filling close to these boundaries especially yes okay okay great thank you I think there is a another question by Andrei to raise his hand please oh the entropy is maximal at the phase transition point between compressible and incompressible phases yes I understand I seem to do with the order of the phase transition I'm just wondering how general this result is uh so I think whenever I mean this is this also proves this that whatever you go from uh from a phase which is incompressible to a compressible phase it does not depend what kind of structuring that you have in the previous phase or the other phase but whenever you go from an incompressible phase to compressible phase you will see this kind of big behavior but that filling should be should be uh so for example in the case of the both last phase or in the case where we do a transition for density say 0.7 for the incommensurate lattice we do not see this behavior so this happens at phase density 1, phase density 0.5, 1.5 so on so forth but for for for the incommensurate lattice case where you try you go from say both glass phase to to a super super fluid phase going across a super glass phase at a fixed density 0.7 you will not see a transition from both glass to super glass but rather see a transition at super glass to super fluid so in that case I would say that it's uh I mean I mean this is not really the the general case general case so I so okay just to summarize uh this would happen whenever you you have a have a fixed density transition for densities that are uh that are commensurately changing from from incompressible to to compressible phase now okay so to answer your question it's not really the the best way to explain if it is a general general scenario for this okay so I mean you'll have to see in some systems um thank you yeah yeah I don't know without introducing both glass phases this might have been a very very um confusing answer but uh in cases where you have uh integer filling like this and non-integer filling like this this would definitely always happen in special cases this might not happen yes thank you are there questions comment uh that I mean I mean these results are for what system like does it depend on system size or like I think this might have this this uh so this big behavior and this will not depend on system size we also see the scaling how this would go with the system size but uh but I am expecting somehow I mean I have a feeling that when this gap would go to zero because of this k is equal to zero mode there might have some special feature happening at super solid super fluid transition so you might see a peak here so in some thermodynamic limit which I cannot really probe so this would be system size dependent yes and you might see much more happening especially at the super solid super fluid transition okay but the general phases will be filled uh yes I mean so you will not obtain any other phase other than these four here okay okay I don't see any other questions so I think we can move to the next speaker