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


So we're in Gdynia, Poland doing the seaport deparcation. So this is the dagger brigade equipment, and we're downloading the vessel today in preparation for our onward movement. Yeah, the dagger brigade is here in support of Atlantic Resolve. We're here to build relationships with our NATO partners, continuing to build our interoperability, our ability to work together. I believe we're stronger together, and so the dagger brigade is looking forward to working with NATO partners, learning from each other as we go about these next nine months. You know, it's amazing to me that only three months ago all the dagger brigade was at Fort Irwin, California, and in the desert, and then now here we are in Northern Poland downloading our equipment, and so in just three months' time we're able to travel so far, project combat power, and I think it's exciting too, the soldiers that are out here to be a part of this operation. It's not something we get to do every day, but they're motivated by it, and they understand the greater and larger purpose of why they're here, and so I think as a team, as a whole, we're looking forward to working with our NATO partners, assuring them and deterring any adversarial aggression against NATO countries. Our main job here is to keep tally of all of their equipment and keep accountability of everything that's being downloaded. We have over 640 pieces and our main purpose is to make sure the equipment is all here, whatever the unit loaded in the vessel all gets here and gets downloaded. We've got Christmas time right around the corner, so a lot of soldiers are going to be away from family, but I think too, this is not the first deployment I've been on, we've been away for the holidays, so I think this is a special time for the soldiers too to understand that we are a family as an army, and so I think it's a good opportunity for us to just come together closer as a team over the next few months.


Hi, this is George Kow and I want to show you how I make social media images. So these are images with a message on it so that it can convey a message on picture heavy platforms such as Instagram and it also is great for Twitter and you could even do this on LinkedIn or Facebook as well. So the first thing I do is go to an image website to download a nice image that would convey the kind of emotion that I want for a message. So let's say that I wanted to talk about the power of purpose or, you know, focusing on our purpose and our everyday work then I go to unsplash.com that's usually where I download images and you could either search for an idea right there in the center and by the way if you don't know what to search for they give you a couple of trending searches you can click on to send to see what's trending right now or you can simply go and scroll. This top part a lot of people don't know is scrollable so just make sure you scroll on the top and just kind of see well spirituality okay let's try that so we'll click on that and then we'll find something that and I just want to let you know the type of image is important is it is it a long image like more of a portrait style this is a portrait type or is it more of a landscape image depends on what you're creating for so if you're doing Instagram Instagram is really a square right so a portrait any of these could work for Instagram you know but if you're doing like a Facebook cover then you want to look for a more of a landscape longer rather sorry wider rather than than taller type of image okay so we're talking about purpose let's say I don't know we're just gonna scroll through and find something that could work for I'm gonna particularly look at Instagram here okay well this person looks like they're they're meditative and thinking so what we do is when we find an image you go and click on it and then you click download okay now it's nice to be able to give credit where it's due so you're posting on Instagram you'll want to just mention mention this in one of the comments if you know Facebook wherever you're putting it try to mention the person's name and unsplash in the comments okay so now the the images now they're on my desktop and let's now go to Canva so Canva is a free design tool and it's it's it's can do a lot but it can also be used very simply to create social media images so log into Canva click on create a design and you can see here that they've already given you a nice sort of you know you can just simply click on this one to start creating an Instagram post or click on this one to create a Facebook cover image I'm gonna create an Instagram post right now so I'm gonna go and do that okay and once that is loaded what I want to do is I want to upload my background image okay so what I do is on the left hand side I click on uploads that's where I go and then I click upload an image and I select the one that is that I've just brought over from unsplash and there you go so once that's one that once it's uploaded I click on it to fill it in to the canvas here all right and let me just move my thing over here oh I'll just keep it here for now make it a little smaller so you can see better okay so once it's on the canvas I want to fill the whole canvas so that the image doesn't have any white space on the and the nice thing is when you click and drag Canva is nice because it'll tell you if you're centered vertically or horizontally you see these purple lines right so I let me go and fill this whole thing up by by basically dragging the the sides and the tops I'm gonna drag the sides and you can see I'm dragging the sides it's an automatically cropping the image so that it fits where I want it to fit so there we go blah blah blah like that okay and I also want to fill up the top here okay so I just want to fill up it just give her the white space and then the next thing I do is I go to elements because if I put words on here it's going to clash right so let's just try try this I'm gonna click on text I click on add a heading that's I don't know why this is so small let me try this again add a heading it should be larger well anyway I can always adjust the font size up here so I'm just gonna say work daily on your purpose keep returning to your focus again and again I don't know just just making something up here and you could see that and I could you could see that the image the words are gonna clash on the image it's gonna be hard to hard to read so what I'm gonna do is I'm gonna put this on this I'm just the way you click and drag an element is you have to move your mouse over the one of the edges one of the sides and then find the sides when it becomes a cross then when it becomes a cross you can you can click and drag it so I'm gonna click and drag this over to the side and what I'll do now is I'll click on elements here and I simply like using either a square or a circle but if you look at the shapes you can click on all and you can see all kinds of shapes that you could add but I'm just going to add a square with a nice round round edge and I'll just go and click on it's free like you can see here Campbell does have some paid elements but I'm using the free stuff for now click on that and they put the element in here now it's covering up the image so doesn't look very good so what I do is I click on this transparency tool up here I click on that and then I would drag the transparency so that I can still see part of the image but then I'll be able to see my words more easily okay so I'm gonna drag the words now and you know maybe maybe I don't want to cover up the person's face so much so well I don't know just for the sake of ease right now I'm just gonna go ahead and cover up the person's face okay I'm gonna drag I'm gonna find the the edge sometimes it's tricky to find the edge oh one thing that's helpful is if I click on this and I click on position I put the element backward a little bit here okay did I do that okay now if I now it's easier to select the the the word element now it's the word elements in front of the the this the square thing here whoops let me try that again move it okay here we go all right I'm gonna center it and I'm gonna make the words larger so 50 something like that and I'm gonna expand the where the size of the words and still hard to read right so what I might do is I might just select a whole thing and make the words maybe white instead a little bit easier to read okay now doesn't look great but you know what you can you can find a better image and you can change the font color and you can even change the element color the element color right now is gray so you know you could try a different different color it's a little bit lighter or blue or whatever whatever you want right purple whatever and then and then you can also make the words bolded you can change the font type there's like dozens and dozens of font types here whatever you want they probably have and and then what I like to do is I like to and this is again I'm not I have I'm doing this in real time so I don't have time right now to make it look pretty but let me just make it a little bit a little bit nicer looking well I don't know maybe a little bit heavier so it's easier to see anyway this is this is not the perfect image but you get you get the idea you can play around with it and then what I also like to do is put a put my website so add a add a subheading okay that's really small but but I will change I'll make sure it's centered right and then I can change the font size to 30 maybe make it wider a little bit again move it so that it's more centered there we go and you know I could change it to white so it's more clear and then here's I can do you know whatever George cowl calm right just as an example so after if I'm finished with that then I simply click download and keep on PNG click download and that's it so I literally in real time without really having a thought about this I designed a social media image for you you know right right on screen and this is what it looks like you know I don't know if you can see this let me let me open the scene here so yeah you could you can see this is what it looks like I can upload it now again I probably maybe maybe choosing a circle element would be prettier would look better but you can again play with different elements here to make it however you like so hopefully that helps hopefully that inspires you and again you could do a Facebook cover image like this too you know Facebook click create a design click Facebook cover image and kind of go to town with that making your own version of your Facebook covers so and they have kind of give you they give you some ideas to start with if you like alright any questions comment below and have fun


So I want to show you guys some of my new property. I guess that people have been asking about So of course, I haven't really moved into a cabin. I've moved into a house. It's a fully fledged house It's around five acres. I might as well show you what what we're working with now So the goal of course long-term Is to start growing a lot of my own food and stuff like that just you know to have a good garden and stuff like that Just so I don't have to worry about You know worry about doing that kind of stuff, you know, I should be doing it myself It's not something too hard so I've started things with that already, but I might as well show you the actual house itself and Well, yeah, we'll start inside because why not? So this house some of you guys know That I've mentioned it to I've probably talked about on streams It's actually been it owned by the family for a long time now. That doesn't mean that I got it for free I actually am buying this but I'm buying it from a More distant relative, I guess like someone a part of the family that's like, you know related by great-great-grand parents or something like that, but Not people I actually know so I am buying it and I'm buying it for what I consider a pretty good price But anyway, so here we are inside now the past week since I've been here I've just basically been decorating. Let me turn on the lights in here because you know, I would if we were like a An expensive channel. We would have already done that We would have had like assistance to do that, but um, so anyway, so the first the main room It's where all the good stuff is it of course originally sort of a mess right now, we'll just go ahead and say that because I Everything like there's so many things I'm putting in place, but so the main room, of course originally is the TV room I got rid of the TV. We don't need a TV and I just replaced it with another bookshelf and chair so I really just have Seating bookshelves.  I sort of was thinking of hanging them as well I've never they've been in like the tube they're delivered in for all this time But anyway dining room got a kitchen. You don't really need to sit the kitchen My master bedroom is sort of at the beginning of the house, but I literally haven't put anything in there I've actually been working pretty profusely since I got here now. I have been doing some I don't know stuff on the computer. This is actually my office by the way Which is a mess as well because I'm moving everything around like where do I want to put my electronics? Where do I want to put my paper stuff? Where do I put blah blah blah? But I have been doing a little stuff on the computer just playing around with stuff As on the seat, that's my usual time waster instead of you know watching Netflix I usually just you know learn to do something on Linux but But this past week instead of doing that I've been doing some of that. I've been like hanging pictures Doing yard work fixing things around the house. I had to fix some doorknobs had to fix Just a couple little things So this will probably be my studio kind of thing I need to put up more of those sound padding things I have a couple of them, but I don't have the sticky stuff that you set them up with so I need to get that Anyway, nothing is too interesting in here You'll probably see more of that because that's probably where I'm recording stuff Although since it is wooden the acoustics are bad So I just put down a bunch of arbitrary rugs that I found just to lessen the the bad sound and sound in here But I'll probably be putting a lot of stuff on the wall as well Nothing my extra bedrooms are back here. Nothing really worth looking at. I do have a garage Which is sort of attached sort of not Got all my tools in there. So let me show you so of course the plan Was to eventually start growing everything for myself. I mean not everything but everything I could you know Now I actually started I'm starting here with a head start now I just started like a compost pile over there that I mean that's not too interesting That's sort of typical stuff people ask me if I'm going to start doing like tutorials on I guess Gardening and if I do do that It'll probably be a little while just because I don't want to give you bad advice because I might be do it like I've already started some things But I don't want to give you advice that like isn't gonna work But I will say there's already a bunch of stuff already here So here are some great plants now. These are not active now in order now I'm doing research on all this while I'm you know sort of starting out here But these are not gonna bear grapes this year just because they haven't been properly pruned and arranged I think you need to you basically need to set you know The whatever vine looks best for the next year and then let them grow so next summer I'll be getting some grapes from these once I fix them all but this summer because you know I just moved in they haven't really been maintained. They're not really gonna produce anything now I do have a couple of fruit bearing trees. I think I have a couple. I think this is a pear tree I'm not quite sure. Let me double-check. I'm not I'm not entirely sure but Whatever it has they're not fully grown a couple of them have fallen Is this a baby pear? I'm not quite sure sort of looks like one But there are a couple fruit bearing trees that have already been here that were planted By relatives that lived here before now where it comes to where I'm planting the garden I've thought there are a couple of areas that are free like that over there I could probably do it over there in the front yard. I could probably do it in the front yard But I'm thinking about actually doing it in the back and that's for multiple reasons One of the issues is sunlight which I probably do get enough sunlight pretty much everywhere. There are some big trees back here But I think compared to the other locations. This might be actually more sunny and I started last week I started making a little fire pit back here. So here's my vision Here's what I want the garden to look like and I don't just want it to be for growing plants I want it to be sort of a nice place generally and You can see there are like hedges That have been planted sort of on the edges of this area the sort of open green area So I'm thinking about using this open green area as a garden and I made a little fire pit In the middle.  I haven't actually used it yet But because I'm thinking about actually making it bigger But you know just get some river rocks just because I like the way they look And so I'm thinking about probably actually making this taller But my idea for the garden is having a sort of a place you can hang out or camp out In here and then a bunch of raised beds around it sort of forming a kind of a maze or just you know Kind of a place that's nice to hang around And I think I have enough sunlight in this area To do that now, of course I could put gardens all over the place Frankly, there's a lot of area to work with In fact, it'd be nice because that's less stuff to mow When it comes to mowing actually the contract that So how I'm paying for this place We have this contract where basically I don't have to do some things in the process of paying for it like mowing the grass is one of them Contrary to popular belief, not all boomers not all 30 year old boomers love mowing grass and I'm I'm not quite 30 yet, but I don't like mowing grass But there is a big mower over here in one of the barns and I also have I also have like a gas canister if I want to get like gasoline delivered You know people usually use it for like farming equipment, but theoretically I could use it for my car Which speaking of which I really need to get I saw you guys might know I have this sedan this Nissan whatever who knows I've been thinking about just getting a pickup truck because Man, there are some things you just read you really need one it like if you if you have A productive hobby life you basically need to pick up because you know, you got a haul Lumber or whatever all this kind of stuff like there've been a million times in the past week That I've needed to pick up So if anyone wants to donate a free pickup to the channel and you live in South Georgia, North Florida hit me up But realistically speaking, I'll probably look into buying one. You know people will have some on sale or whatever Let's see what else there is So there is this little shed. I sort of walked by it when I was looking at this pear tree But the shed it Has a nice little well In it. I mean, that's where my water comes from and comes from well I am on electric like Public electric or whatever, but that's pretty much my only bill I guess I I do have the internet as well But I'm thinking about doing internet here is pretty bad, but I have been able to upload videos, especially if I've uploaded this one, but Anyway, so this kind of shed thing I'm thinking about I think one of my relatives. Oh look an old dog bone Anyway, one of my old relatives actually did have a dog that he kept In this shed So there's this shed and he had a little opening where it could come out and play around And that's important because like out here. It's so wild that If you have pets, they're going to get eaten by something eventually, you know coyotes or something coyotes are all over the place I think there are boars But I'm thinking about doing something similar As he has done, you know, sort of make a little fence out here and then make a little door for like chickens or something to live in That's sort of how I'm thinking, you know having You know Chickens or something else just sort of have an area for them and it's right next to where I'm building my compost file Okay, so let me think is there anything else So that's a lot of the basics. I'll probably do more videos on more specific stuff. So I don't want to get inside It's actually it's not that hot out today But uh, I don't know. I think it's when you're walking around and talking It just takes a lot out of you takes takes, you know, because I don't ever take breaths in between talking So, oh, it's so nice and cool Oh, of course, I'll probably get my first electric bill here And it'll be like way more than I want to pay and I'll like never use the ac again But I'm just going to enjoy it for this month before I see my bill Um, anyway, let's see what else is there to do. Maybe I'll show you some pictures of myself and that'll be the end of this video Um, let me show you chronological pictures.  It's about what I looked like In first grade I got glasses. Let me see if I can find a picture like actually, I guess this one right here There's me and my brother. I'm the one with glasses and uh Here is one. I want to say this is a middle school. Look at this smug kid. Yeah, that was me That was me kids Uh and went in high school Over here. This is what I look like. Look at this edgy kid out the edge Yep, that was me in high school But then like during my 20s, especially once I moved out of the house There just aren't many pictures of me or maybe there are a lot of pictures, but since I never had facebook I never saw any of them. Um Let's see. Yeah, I don't know if there are any others in here Most of these pictures I got most of these from my dad's storage shed I think I said before but uh a lot of them were in the storage shed because they're like really old Um, I would show you all these pictures whether there are people aside from me So, you know, I don't know. It doesn't matter that much, but um Anyway, so that's about it. I guess um, so I'm definitely enjoying this place Let me turn off the light so I can save electricity I'm definitely enjoying the place And uh, I got a whole bunch of plans for it, but we'll see how it goes. I um I guess that's about it So anyway, if you have any suggestions any kind of garden suggestions or anything else Feel free to hit me up Otherwise, I'm probably going to be hanging pictures for the rest of today Probably hanging stuff in that office and maybe in my uh my dining room here I just to make the walls the walls in here are looking a little better because they have stuff all over the place But everywhere else is a little bare so I gotta fix that Um, so anyway, that's about it and I'll see you guys next time


So here's our second example question. So we've got a Voltex cell constructed from a half cell which cadmium. So I'm going to highlight some important bits as we go through. So we've got a cadmium rod dipping into cadmium nitrate. So that's our electrolyte on that side. On the other half we've got a silver rod and that's going to silver nitrate. So that's important information to know. So we've got what our metals are and our electrolytes. Two half cells connected by a salt bridge so that's good to know. And the silver ions are being reduced. That is very important. So the silver is being reduced that means that the cadmium has to be oxidized. So draw a sketch to the cell, label anode and cathode, write half reactions, indicate electron flow and sign of the electrodes. So let's do that now. So here's my diagram. I've hooked up a voltmeter. So first bit of information that we were told that's very important is the silver ions are reduced. Now I like to have the oxidation half cell on this side and the reduction half cell on this side. So I'm just going to label that quickly. So that side is going to be ox and that side is going to be red. So the silver ions are reduced. So if we look back at our information we're told that there's a silver nitrate is our solution down here. So we've got our electrolyte is agn03 and on the other side we had cadmium nitrate. Okay now silver is being reduced. So we know the site of reduction is the cathode. So this is our cathode and consequently the cadmium is being oxidized. So that's our anode over here. We're going to get a build up of electrons down here as the cadmium is oxidized. So a build up of electrons is your negative charge. So the anode is going to have a negative charge. The cathode is going to have a positive charge. What else do we need to know? Let's look back at our question. We'll do the half of the occasions in a second, direction of electron flow. So build up of negative charge on this side so the electrons are going to be pushed away. So our electrons are going to move from the anode to the cathode like that. And now we can write our half equations. So our cadmium rod, so the rod there is made out of cadmium. Over here we have our silver rod. The cadmium is being oxidized. So oxidation is loss. So cadmium, so solid cadmium, that's going to cadmium ions. CD2 plus and that's aqueous and plus two electrons. So it's losing the electrons. Over here the silver is the site of reduction. So the silver ions are being reduced from the solution. So AG plus aqueous, that's dissolved in the solution. Plus one electron because silver only has a charge of plus one goes to solid silver, so AGS. If we wanted to we could then write the overall equation by times this one down here by two. So we have the same number of electrons in both sides. And then we'd be cancelling out the electrons. So just write that out quickly. So we've got 2 AG plus aqueous, so it's two silver ions in solution plus two electrons goes to two solid silvers. Now we can write the overall equation if we like. So we can set out the electrons which are on both sides of the arrow. So we've got cadmium solid plus two silver ions in solution goes to cadmium ions in solution plus two solid silvers. All right. So that's an example of electrochemical questions. That's it for the video. Thanks for watching. I'll see you in the next video. Bye-bye.