 Good morning to everyone. It's my pleasure to host, to chair the session about Gaples or long time correlations and our first speaker is Arithra Kundo from CISA, who is going to tell us about revisiting Mazu Suzuki bound for a finite system. Actually, the program changed and I switched with Arithra, so I am the first. Oh, okay. Ah, yes, you have not looked at the last version of the program. It's okay. I'm sorry. Okay, so the first speaker is Stephanos Gopa about talking, also from CISA talking about correlation functions of interacting 1D and unique gases. All right, so Stephanos, the floor is yours. Yeah. Can you see the screen? Fine. Yes, we do. Okay, so good morning to everyone. Thanks for the invitation, for giving me the opportunity of presenting my work. I'm also happy to be the one who opens this Adriatic Conference on strongly correlated systems. And today I will talk about correlation functions of interacting the one dimensional ionic gases. So, okay, I cannot go on. I have a problem. Sorry. Okay. Okay. So, my presentation is divided in two main parts. The first one is an introduction, where I will present the ionic statistics in one dimension and the Kundo model, which is a model for point wise interacting anions and which has an exact solution based on beta answers. So, I'm going to review briefly the better solution and its thermodynamic limit in the in the main part of the seminar I will focus as you may have guessed from the title on correlation function and specifically I will consider the asymptotic behavior of the one particle density matrix. Next, I will provide you a pedagogical introduction to bosonization and a lack of liquid techniques applied to anions, and also some technicalities regarding the computation of non universal amplitudes that are involved in this correlation. The message that I would like to leave you is understanding how the unique statistics modify the behavior of the correlation. And also to answer the question of what happens in the presence of a confining potential that's conferred to our onionic gas, the non trivial density profile. And then I will jump to conclusion. So, first of all, I should probably answer the question, what are anionic particles. Well, the simplest way of understanding the ionic, ionic statistics is perhaps to consider the particle exchange operation. And it is well known that if you consider the exchange of two identical boson, the wave function remains unchanged. And if you do the same with permanent particles, then you get an extra minus time. But now imagine that you have a third quantum realm where you have anions defined as under particle exchange, they can get any phase, which can be anything between zero and five. This non trivial phase is typically parameterized in term of an ionic parameter couple that takes values between zero and one, and that has a special limit for couple equal to zero, the bosonic statistics and for couple equal to one, the fermionic, the fermionic statistics. Now we can rephrase this simple idea about anionic statistics in a second quantization language by introducing a set of one dimensional ionic fields, satisfying some generalized commutation relation that depends continuously on the values of couple, and that interpolate between the usual bosonic and fermionic statistics on varying of couple. Now the natural question that may arise is, are these anions something physical? Well, I have a problem with the change of slides, sorry. But to answer this question, let me consider a very simple model where we have a charged particle, which is orbiting around the magnetic flux. And by the way, this example has also an historical relevance. Now in this setting, the particle is a subject to the presence of an azimuthal vector potential, and it is known from standard quantum mechanics that the dependence of this vectorial potential on the dynamics of the particle can be absorbed with a gauge transformation. In the gauge field, the wave function gets a non-trivial phase factor and obey a free evolution with a non-trivial boundary condition. Or in other words, you may look at this problem as if the orbital angular momentum eigenvalues are not just given by integer, but they are shifted by a certain amount delta, which depends on the charge q and on the magnetic flux p. Now if you consider not just a single particle, but a system of two identical particles, it is easy to see that a similar boundary condition is satisfied by the two body wave function. If we don't consider now a full rotation, so a two-pile rotation, but we restrict ourselves to a pi rotation, we can see by construction that the two body wave functions still get a non-trivial phase. And then, I mean, it's also easy to see that the pi rotation corresponds to a particle exchange. So the composite system of charged particle and magnetic flux effectively behaves as an anionic excitation with anionic parameter kappa, which is given by delta divided by 2. Let me also mention that spin statistic theorem, the anionic statistic is compatible with the spin statistic theorem since the latter holds true only in three dimensions. And for d equal to 2, the exchange operation is no longer compatible with a simple permutation group, but particles can also braid and so fractional statistics are allowed. Let me also mention two remarkable examples. The first is the fractional fundamental effect where we have a two-dimensional conductor kept at very low temperature and on this conductor will load a gas of electrons subjected to a magnetic field. Now, if we measure the whole conductance in this setting, we observe the anomalous series of plateaus and these plateaus were interpreted by Lowling as quasi-particle excitation of anionic nature that carry a fractional amount of the elementary electric charge. There is also a nice application on anions which regard topological quantum computers where you consider a system of anionic particles and you associate with the braiding of these anions a certain logic gate. In this way, if you look at the phase, at the topological phase that is accumulated over time in the system, you can associate in this phase a certain amount of quantum information. And this quantum information is stored in a non-local way, being actually much less fragile against the coherence than the standard use of qubits. But as you may have seen, my example refers only to two spatial dimensions, and my talk is about D equal to one. And there is this common belief to associate anionic behavior only in the two spatial dimensions as it is confirmed by the real first statement that you can find in the Wikipedia page of Anion. So maybe the question should change to, are anions something that is observed only in two dimensions, or how could we think about anions in just one D? Well, let me try to address this problem. First of all, let me say that in one dimension there are bosonic models with certain Hamiltonian as the one that I'm showing here that can show effectively anionic behavior. But this is not the point of view that I'm going to follow during my presentation. I would consider anion not as some emergent degrees of freedom, but as a starting point for theoretical interest. And I would like to understand how does the anionic statistic modify in an object or if there are new features that can be investigated with anion. And this point of view is not just for theoretical interest, because there are some, there is the possibility of experimental realization of one dimensional anion. For instance, in 2011, tailman and co-authors managed to realize a one dimensional anionic model. The idea is to consider an anionic upper model, which is the upper model just made with anions that here I wrote as this field A, and then to realize that an anionic upper model can be written simply as a bosonic Hamiltonian with an occupation dependent topping that can be tuned with Raman's Petroscopy. And in this way, in experiments, you can achieve a fully tunable particle exchange statistics. And let me also say that this experiment was made at finite interaction and in the presence of a confining potential. And both these constraints will be will be considered in the derivation of my results. In the only instance, there are other studies as the one of Grazner of 2015, where they improved this setup or also other techniques based on Flokke force in the engineering as the one of rather of 2016. Okay, so somehow we we had an understanding about anions. Now we should confer to this anionic gas that certain interaction. Of course, you can do this in in many possible way, but perhaps the simplest instance is to load our gas on a ring and to confer them a point wise repulsive interaction. Now in a similar way, as you do with bosonic model considering the Liblinigar Hamiltonian. And in fact, what you get is exactly the same Hamiltonian of a Liblinigar model, except for the fact that now the fields are replaced with anionic field. This model is known as anionic Liblinigar model or Kuhn's model. And it is also interesting to note that as the Liblinigar model provided a set of periodic boundary condition can be sold exactly by means of beta answers. The Kuhn model provided a set of boundary condition that are compatible with anionic statistics can be sold as well with the use of beta. After reviewing this briefly this beta answer solution, let me just say that this Kuhn model emerge as a special limit of the anionic subbar model and in particular when we consider low temperature and low feeling in the same in the same way as their bosonic counterparts do. And so we can say that at the actual state of the art with experiments, the Kuhn model that will be the main character of my thought can be realized. Okay, let me just give you a sketch of the beta answer solution. Basically with beta answers we made an hypothesis on the structure of the eigenstates and specifically we parameterized the eigenstates of the model to be a certain function of some spectral parameters called rapidities. And then we impose that this wave function these eigenstates satisfy the associated quantum mechanical problem in first quantization and the right type of boundary condition. If you do that, then you end up with a set of equation called beta answers equation from which you can, you can, you can get the values of the rapidities and so the eigenstates of the model, but not only also all the conserved quantities of the models such as the energy, the momentum, again values and so on. Now in this equation, these ij are called beta integers, they are either integers, either same integers, which are in one to one correspondence with a given eigenstate. For concreteness, in the case of a ground state, these beta integer are equally spaced numbers between minus n minus one half and n minus one half. And this C prime also that appears here is an effective coupling, is the coupling constant of the gas divided by a certain factor caused by kappa over two, which depends on the anionic study. Now, who of you that are familiar with beta answers, you may have recognized that these beta answers equation looks almost the same of a Libri-Niger gas, except for this C prime. The only difference is contained in this orange term, which depends on the on the anionic statistic, and this orange term may be viewed as a topological shift of the rapidities in a somehow a revolutions of an Arab bomb. Let me also consider the thermodynamic limit, and in particular I'm going to focus on the ground state properties of the model, and I'm going to take the limit where n and l are large at fixed density. When we consider the thermodynamic limit, the rapidity becomes a continuous variable, so it is useful to think no more in terms of rapidity, but in terms of a distribution of rapidities called the root density. This root density satisfies an integral equation, and where this Q here is typically called the Fermi point and can be fixed via the value of the density. Now, from this equation, from this integral equation, we can clearly see that the rapidity shift due to the kappa dependence, this appeared in the thermodynamic limit as a sub leading contribution. And also, again, for who of you that are familiar with the better answers, this integral equation is the Libbe equation. So the equation that you would have get if you you you started from from a simple bosonically linear model, except for the replacement of the interaction C that goes into C prime. So a question that we can ask is, is everything the same if I consider the Pund model at t equals to zero and in the thermodynamic limit to a bosonically bleeding model with the same with a simple change of the of the interaction coupling. And the answer is yes, if you focus only on the spectral properties of the model and under this, this, this assumption, you get exactly the same spectral properties of the Libbe linear model. And I also mentioned that you get the same generalized hydrodynamics. And this is actually not surprising because we expect to see some effects related to dynamic statistics, not when we focus on one point object, but when we consider quantities that depend on more than what, than, than, than one point. So when we consider correlation function for correlation function, there are new features and I'm, I'm just mentioning that the strong repulsion limit was already discussed in literature by many people. And I'm quoting here Pasquale Calabrese, Mihail Mincev, Raul Santa Chiara, and many others. Instead, the finite interaction case and the case of a non homogeneous gas were addressed for the first time in, in our world. So now we should compute this correlation function basically we have two ways in which this can be done. The, the first is a direct approach where you consider the beta against states you follow everything in and you try to compute the multiple integral. However, these approaches are the, because the complexity of the beta wave function grows really fast with the number of particles and so in practice is the limit stated just to a bunch of particles. In alternative, what we can do what we are going to do is to focus on the low energy asymptotic behavior of this correlation function using a lot in generally with you. To prepare the ground for that interliquid theory, let me first bosonize our onionic model. Now, it is known, it is well known that in one dimension, every any kind of fields can be expressed in terms of a bosonic field that seems statistical and scattering phase can be thought to be on the same footing. And doing so basically our onionic fields becomes a bosonic fields time a certain stream. Now, you are probably very familiar with the Jordan transformation between the boson and the spin less fermions when it comes to onion you just have to add a couple of years. So, in terms of these of these bosonic fields that we construct a low energy description, considering the harmonic expansion basically we take a density phase representation of the of the bosonic field. And then we focus on on on the at low energy to large scale fluctuation that that are dominant if you consider low momentum process. This large scale fluctuation can be can be written as a background, the density that I called here just pro plus a fluctuation field that I call the capital pi that encodes this density fluctuation over a background. And that they correspond in the microscopic model to a particle or excitation that is formed at the Fermi point. However, this dive gas is not capturing the the world behavior at low energy of the model because there are also some types of excitation that carries a finite amount of momentum. And you can think for instance to a back scattering process or to a new class excitation. So, the idea now is to consider to to unveil the discrete nature of the fluid by considering a leveling field that is defined to be a certain function that undergoes the pilot continuity at each position where a particle is found from the left to the right of the chain. In terms of this labeling field we can not just consider the the large scale fluctuation but also read the shorter wavelength contribution. In the end you arrive to this expression for the density operator from which you can you can extract the right behavior at low energy for the bosonic field, which is typically called harmonic expansion. And then you just have to consider to take back your your your statistical phase and you arrive to this result. Now, for you that are familiar again with bosonization, this is the result appears really natural. And this is the sum is an infinite sum of a integer. And actually it is it is well known that for bosons you some only on even integers, whereas when you consider fermions you some only on all the integers. So since the aliens are placed between these two limiting cases it is it is natural to expect that you have this sort of behavior. And let me also comment that is this amplitude here because of them is a non universal amplitude that arise that contains all the degrees of freedom that have been traced out when we we go when we taken when we have taken the the limit of the field theory. But we will we will fix them in a moment. So, we we expect our, our anionic fields in terms of density and a phase degrees of freedom, and then there are the essence of Lattinger liquid is too bright for these for these fields, a quadratic coming from which is the last in terms of liquid in terms of summit on this emiltonian depends only on two dimensional on two parameters, the sound velocity and the last integer parameter. These parameters must be fixed from the microscopic model so from beta. say that the way in which you can fix these parameters is exactly the same that you do with a Liblinigar gas, except that they depend not now on the reduced coupling C divided by rho, but on this effective coupling gamma prime, which is C prime divided by rho. And here for completeness, I am showing the results for the last integer parameter as a function of gamma prime. The last ingredient that we need is then to promote our last integer liquid, Hamiltonian, to a one plus one dimensional conformal field theory. And doing so, we get the conformal field theory of a free massless boson. Why this is useful? Because if you want to compute correlation function, and in particular, I'm going to focus on the one particle that's ematic, then the expectation values of the anionic field translates into the expectation value of some objects called in the CFD language vertex operator, whose expectation value is known for different types of boundary conditions. So, replacing the computation of this correlation function into a CFD problem, we can get the correlation function with almost zero computation. And so, we may say, okay, this is the end of the story. Well, not really, because this expression is formally correct, but there are two issues that must be fixed. The first is that this sum is an infinite sum over all the integer. And so in practice, it doesn't have a real predictive power. And the second is that we don't, we still don't know the structure of this non-universal amplitude. So, we should fix that. So, let me try to address both this point. So, the first thing is to notice that this non-universal amplitude has a dimension, and it's always convenient to work with dimensionless parameters. So, I can just extract from the amplitude its scaling dimension that are given by the vertex operator. Then the idea is to retain only the leading order or the behavior so that we can truncate our infinite series. And by ordering the harmonics in power, in terms with increasing the scaling dimension. However, as you can see, the scaling dimension of the vertex operator depends explicitly on CAP. In particular, if you consider CAP equal to zero, we have the botonic case where it is known that the leading harmonic is the one m equal to zero. And we have choose our leading contribution, the one m equal to one and minus one with equal weight. But when we turn on CAP, so we consider an anionic statistic, we observe that the harmonic m equal to minus one contributes more and more. And eventually it becomes a leading order when we move to the fermionic point. So, the idea is yes to keep the leading order, but we should keep in mind that the leading order is given by m equal to zero when CAP is different from one. But when we approach the fermionic point, we should keep also the term m equal to minus one. Also, at the fermionic point, by exchange symmetry, it is expected that the two leading amplitude that are involved here are of the same order. So, in the end of the day, we should fix only one one leading order amplitude that I called small b zero of CAP. But how to do so? Well, let me say that there are some analytical results for this object in the limit of strong repulsive interaction. This computation was based on a Fisher-Rafting conjecture and was done by Santa Chiara and Calabria. In contrast, that finite interaction, nothing was known, and so we should go back to the microscopic model so to better understand. The way in which we fix this coefficient is given by this formula, which may look complicated, but it is actually easy to understand. We are relating the non-universal amplitude to the expectation value of the anionic field between two states. These states, given in terms of this rapidity lambda, identify the ground states of the microscopic model, which correspond in the field theory language to the value. The other state, given in terms of another set of rapidities, mu, is an excited state of the microscopic model, which is in correspondence via operator state mapping to the vertex operator of the CFD. Once we identify this state properly, we can relate this object, which is called in the beta-antus language form factor, with the proper normalization to our coefficient. Note also that whether in the microscopic model this object can be computed only when N and L are fixed. We consider N and L to be large, but we should fix their value. Then the thermodynamic limit is taken just considering at large n and 1 over n numerical extrapolation. The result that we got is summarized in this plot, where you have the leading order amplitude for different values of the anionic parameter and the unbearing of gamma prime. We properly reproduced the bosonic behavior when top power is equal to zero, that was known. Also, we reproduced the analytical results of Calabria's Santa Chiara when gamma is large when we are in the strong repulsion limit. Just a technicality, as I said before, this object, similarly to the wall correlation function, is really hard to compute if you just plug inside this object the form of the beta-age state. To get to big values of n, let's say I don't know, 20, we needed a determinant formula for field form factor and this determinant formula for the Kundo model wasn't known and it was a byproduct of our study. So now that we have all the ingredients, let me show you the plot, the result for the one particle density matrix. Here I'm considering the periodic boundary condition and we have the real and the imaginary part. Each of the of the panel shows three values of the interaction coupling for c equal to 1, 10 and 100 and different rows show different values of the anionic parameter. We can see that the system develops oscillation of increasing frequency when we increase the value of kappa and when kappa is close to 1, we observe a beating effect due to the superposition of the two harmonics that we consider. Also, there are quantitative differences with respect to the strong repulsion limit and the correlation at large distances appear to be announced. But I would say that the best way of understanding the differences between bosons and anions comes out when you don't consider the one particle density matrix but you rather consider its momentum distribution function which is given by the Fourier transform of the one particle density matrix. Now, if we consider bosonic system, the momentum distribution function is a symmetric object with a peak located at q equal to zero and with a certain a that is a power law dependent depending on the last integer parameter. When we turn on kappa, this symmetric momentum distribution function starts to be asymmetric and in particular the peak initially observed at q equal to zero is dragged backwards to a position which is proportional to minus the anionic parameter kappa and its height decreases also with a certain power law that now depends explicitly on the value of kappa. When we arrive to the fermionic limit, this singularity forms the singularity which is observed at minus the Fermi point. Instead, the other discontinuity is formed by weaker singularities located at position 2 minus q times the Fermi point which becomes of much more stronger when kappa converges towards 1 and it becomes that kappa equal to 1 of leading order. So, somehow we established the behavior of correlation function as finite interaction. Now, we want to understand what happens in the presence of a confining potential also because this kind of question are useful for a possible experimental realization of the model. So, we consider the kundu model and now we have the term dependent on the on the trap. However, if you do so, any anomogeneity that you introduce in the model breaks down the variance of solvability. However, nevertheless, we can assume a scale separation hypothesis. What does it mean? Well, basically it means that you consider a very smooth trap that is a very slowly very in function of space and you focus on a small segment of the system, a fluid cell, where by local density approximation the trap looks almost flat. So, you don't have an anomogeneity, so you can approximate your potential to be just a con. And in scale separation hypothesis, we assume, we further assume that on this scale, the system accounts already a thermodynamically relevant number of particles. So, practically this means that we consider the thermodynamic limit of the betanus equation and the local density approximation in a consistent way. Under this assumption, basically everything boils down to run the thermodynamic betanus algorithm for each of the fluid cells. And also the description of the low energy properties of our anionic models remains almost unchanged. For the computational correlation function, we will need some extra tools that I'm going to tell you soon. Let me just briefly review the anomogeneus Latinxer liquid, where with the same strategy of before, you get a description of low energy for the field and for the Latinxer liquid amine zone. Here, of course, the density now depends on a fluid cell index x, but also this happens for the non-universal amplitude and for the velocity of sound and the Latinxer param. However, this dependence is the simple dependence on the density, on the local density, through the reduced coupling, the gamma prime, which now depends on x. And so this can be computed in the same way which we did for the homogeneous case, just considering at each time a single fluid cell. The difficulty instead comes when we consider the CFT, because now the CFT is the CFT of a boson, but with a non-flat magic, which is a consequence of an especially dependent propagation velocity that treats the spatial and the auxiliary time coordinate in not in the same way. And this is reflected in the one particle density matrix, since now our expectation values of vertex operators is computed over a non-flat space. However, okay, there is a trick to overcome this issue, which is a vial transformation. Basically, we look for a change of coordinate of the spatial coordinate x that goes into this x field, such that the line element that is associated to this non-flat magic is proportional to a flat matrix 10. And it's easy to see that this simple transformation does this job. If you do that, then since the vertex operator behaves as a primary field of the CFT, what you do is just to pick up a factor on top of an overall factor for each of the vertex operators that are involved. Now, it is also interesting to visualize this transformation x that goes into x field, and in particular, x field is a stretched coordinate. Now, if we have a non-flat especially dependent propagation velocity, ds, for instance, of this shape, we can instead of considering the spatial coordinate, we can consider the time that is needed by an excitation which is emitted at one border to reach a certain position. Of course, if you consider positions that are very close to the border, where the velocity is close to zero, you need a really large amount of time to move towards higher values of x. Instead, if you are close to the middle of the trap, the time needed is much smaller, because the velocity is large and so on. Okay, so somehow we understood that this issue about especially dependent propagation velocity can be managed as switching the space into a traveling time of excitation. And if you plug everything inside the one particle density matrix, now you have your expectation value, which is computed with this stretched coordinate. However, this change of coordinate is removing the dependence on the velocity of sound on this non-flat tensor, but it's not removing the dependence on a specially dependent Latinxer parameter. And I also mentioned that this dependence cannot be removed with a further change of coordinate. And so just rephrasing this situation, we have now an expectation value which is computed over a flat space, but with a non-uniform medium. And in this situation, there are no boundary CFT methods that can be applied if we exclude limiting the cases as the one of strong repulsion. Nevertheless, we know that our Latinxer liquid Hamiltonian is a pre-theory. So in the end of the day, we need only the expectation values of these propagators, of this two-point function, and then more involved objects as the one we were looking at can be obtained with the use of this theorem. A possibility of computing this two-point function can be to put our problem into an analogy with Coulomb gas formalism to a two-dimensional classical electrostatic problem when one is looking to the Green's function of a system which is firmed by a non-uniform dielectric problem. If you do this analogy, and I will keep this story short, you will end up with an equation that defines your correlation function as a Green function. And this equation, mathematically speaking, is the Green function of a generalized Laplace operator, but also it can be looked at as a certain Maxwell equation for the derivative of these objects. This is, for instance, the one for fee-feed and, of course, the other. And if you do this job and you numerically evaluate this Green's function by inverting the kernel, you obtain these objects that now the theorem I'm showing for this coordinate R, which is a simple normalized structured coordinate. And in particular here, I'm showing the results for the strong repulsion limits, because there we have an exact solution that we can, we use for a test, for a non-trivial test of our code. There are also other techniques as the one developed last year by Alviso Bastianello and other encoders, which is very efficient and gives access to the whole correlation matrix. It's also to the off-diagonal element of the correlation matrix as this one that we need for the computation of our correlation function. Nevertheless, apart from technicalities, if you get numerically these objects, you can plug everything in and you can explicitly have the results for the correlation function. Basically, combining everything together, we can show the one particle that's in the matrix, again, the real and imaginary part for different types of confinement. Here, I'm showing the results for an harmonic trap, a double well trap, and a triple well trap. This is the density profile. The different curves refers to different values of cap, and let's comment a bit these figures. Basically, all the the properties that I've discussed for the homogeneous case like oscillation and so on, still are still present in the case of a trap. But also on top of this, we observed a marked spatial dependence on the type of the trap that we consider. And this is a spatial dependence, as I mentioned, that is much less evident if you consider large values of the interaction. Also, let me say that we reproduced for kappa equals to zero the Bosonic case that was investigated previously by Yanis Brown and Jerome Dubai, and that was test also against the MRG simulation. And last comment, in this case, the non-universal amplitudes are actually function of the fluid cell of the spatial coordinate. So not even a qualitative estimation of this correlation function in the non-homogeneous case were possible before since this non-universal estimation of this non-universal amplitude weren't known. So maybe I spend just a minute for the conclusion. Briefly, we consider this kundu model, and we employed a Latinger liquid description for its asymptotic correlation function with a particular focus to the one particle density matrix at finite interaction. To do so, we needed and we obtained a determinant formula for field for factors of the kundu model. And using non-CFT techniques and tools coming from recent literature on a non-homogeneous Latinger liquid, we were able to derive and explicitly plot the one particle density matrix at finite interaction and also in the case of non-homogeneous gas. Surely what is next is to investigate the non-on equilibrium dynamics, where the right evolution of the one particle density matrix requires the use of quantum generalized hydrodynamics that accounts properly for the fluctuation of the Fermi kundu. This is interesting because there are some hints that suggest that possible dynamical fermionization and bosonization of the anionic cloud after a trap release protocol, as it was evident by a preliminary investigation of the anionic tungse girardolimit, so strong repulsion, but the interacting cases are still under analysis. If you are interested, we published two paper on the subject. One is on JCCA and one is on Gestat. And let me also give a big thanks to my collaborators, Lorenzo Pirolli and Pasquale Calbris. And I also thank you all for your attention. Thank you, Stefano, for this night talk. I clap you on behalf of all of the audience. And with that, we can start our discussion and questions. I see that there are no questions at the moment in the questions and answer section, so please go ahead if you have any questions. So maybe while people are writing the question already in their hands, can I ask a question? Yeah, sure. The problem is that I cannot see any more at the bottom of the Zoom application where there is a raised hand. But don't worry. If we have questions or raise hands, we will tell you. Okay, thank you. Thank you. Yeah, please. So my question is, can you re-explain please, because I might have missed this part. How you introduce the innomogeneity in your model, that is to say how you come from the lattice model to the inhomogeneous continuous model. Yeah, you probably replied to this part. Oops, this part, right? I believe so, yes. Yeah. Now, okay, the idea is very simple. If you just take your microscopic model, so your kundu model, and you try to solve it with beta answers, actually you rely on a model that doesn't have any innomogeneities, because as soon as you introduce innomogeneity and beta answer solvable model, you break down its solvability by construction. So in principle, this model cannot be solved with tools from beta answers. However, since we are interested in a larger scale, in an asymptotic large scale behavior of the correlation, what you can do is to imagine that your potential is very, very smooth. It's a slowly varying function of space. And then if you focus on a small sub-segment of this system, basically the effect of this potential is simply a constant. And so this problem by itself can be solved with beta answers, okay, under simple LDA approximation. The problem, and that is why we call this, I mean, it's not my term, it's a term that comes out in the recent literature, why we call this scale separation hypothesis. Because naively, if you do that, basically you, under LDA, you don't have a thermodynamical relevant number of particles. So you end up with a microscopic model where this potential is flat, okay. The scale separation hypothesis on the other hand tells you that the scale over which the microscopic model can be interpreted as counting a large number of particles, so it is in the thermodynamic limit, is separated from the scale where the potential looks flat. So if you place you in between these two scenarios, so you investigate the system at this intermediate scale, where the system is in the thermodynamic limit, but also LDA can be applied, you can both have results from thermodynamic beta answers and the use of LDA that allows you to remove the homogeneity system out, okay, that's the idea. I see. But so you don't use a particular conformal field theory formulation? Actually, okay, okay, actually, this is a nice question. The point, okay, in principle you can use, with this scale separation hypothesis, a CFT formulation in a in a fluid cell. The problem that doesn't allow you to use the CFT is the fact that when you end up with an action that has a spatial dependence here, okay, so here you have two special dependents, the one on the metric, but we have seen that you can remove it. So in the end of the day, you have these dependents on the Lattinger parameter. You cannot use the CFT because you have this guy that can be interpreted as a especially dependent mass of the field theory. But this is, in my case, I wasn't able to use the CFT to get an analytical result. But there are cases, for instance, when you consider a stronger portion, this Lattinger parameter, no matter the position of the system where you are, it's identically equal to one, okay, this is the result of the Lattinger parameter for three fermions. And since you don't have in the end of the day this k of x, because it's just one, you can go on with the boundary CFT and you arrive to a fully analytical result that actually we consider it in the paper, but okay, I'm not reporting it this year. So the real problem about CFT is not the scale separation hypothesis, it is the presence of this Lattinger parameter. But actually I'm a bit surprised by one of the sentences you just said to say that when it's a free fermion, but with the space dependence, there is no problem to write the CFT, but in that case, aren't the holographic and anti-holographic coordinates dependent on the position themselves? And as a consequence, the CFT that you're writing is no more a CFT because of the dependency of the coordinate themselves? Ah, yeah, but of course, I mean, I'm speaking about the CFT that you write in terms of this stretched coordinate. Sorry, yeah, yeah, okay, this is, yeah, you are absolutely right, but after the change of coordinates to the stretched coordinate, your CFT is defined in a two-dimensional space where the spatial coordinate is now this stretched coordinate. So remove the problem you were talking about, and then you have an auxiliary time coordinate. That's the trick, yeah. So I read, I see that there is a question. Yes. In one dimensional system, you cannot have braiding, right? Yes, you cannot. In fact, I mean, my introduction was just about anions in general. Okay, this refers only to two-dimensional systems, and also this part refers to two-dimensional systems. What we do as I tried to point out is that we just consider anionic statistics, in particular, these are abelian anions as a starting point. So we just assume that this commutation relation are true words. And yeah, you don't have this braiding aspect that you are talking about that is just in two dimensions. This is my starting point, okay. From this commutation relation, I am assuming that you have this set of fields and then I derive my results. But yeah, and why it is possible to do that? Well, for theoretical interest, but also you can realize this model. So if you get something interesting, there is always the possibility of doing, engineering this one-dimensional anions in an experiment. And also, okay, but this is not really the point of view that I prefer the most. There are one-dimensional models, bosonic models, that with some strange Amazonians, like this generalized delta-amazonian, they have as an emergent and effective degrees of freedom, this quasi-particle of anionic nature excitation. So yeah, but anyway, yeah, it is true. We don't have braiding in 1D. And maybe I'll go back to the last slide. Okay, so if no one is asking the question, I also have one question. So this mapping between anions and bosons in 1D, which was shown in the slide a second before. So are these ordinary bosons without any constraints on the local hybrid space? Sorry, I cannot hear you very well. Can you just say that again? So this mapping between anions and bosons in 1D. So are those just ordinary bosons with... Yeah, I mean, yes, you are probably referring to this part. Let me go back. Yeah, this part, right? Yeah, yeah, basically, just that you have a certain statistics, okay, the anionic one, and then you express the anionic field. Here I put a label zero, just because in my convention, the field depends on kappa as a parameter and kappa equal to zero is a simple bosonic system. Yeah, yeah. So you have bosonization with this string and then a standard bosonic field. That's why I'm using all the aldane construction for the low energy description of the field. Yeah. Yeah, it's a standard boson. And you can understand this also because if you go back to... Okay, it's not super... It's not something that you can... Maybe I was too fast, but if you plug the zero here, basically, this relation reduced to the standard bosonic one. You really interpolate between standard one-dimensional bosons and spinless fermions with this construction. I don't know if I answered your question. Yes, thank you. Yeah. Yeah, so actually, I have... So could you... We have still some time. Uh-huh. Could you say something about this thermodynamic better answers algorithm? How does it work? Does it start from these better equations, which are equations for rapidities? Ah, yeah, yeah. Okay. I don't have a slide on this. I'm sorry, but I can try to explain you. Basically, in the thermodynamic limit... Okay. Okay. Basically, in the thermodynamic limit, more or less, you have to solve this equation. Okay. This is a question for the root density, from which you can reconstruct other quantities that are interested for the model. Well, this... At a t equal to zero, since this integral equation is bounded, you have this Fermi point. Okay. So this integral equation is actually bounded. You can just discretize everything, and then this integral becomes a certain kernel, a Riemann sum, but then you can express everything in a matrix language, so it's a kernel. And then you just make a kernel inversion, and it's really efficient and easy to get this root to that. When t is not equal to zero, it's a bit more complicated because this q becomes infinite because thermal excitation can... I mean, you can have particle outside of the Fermi sphere, no? And so, yeah, this q is no longer finite, and so you rely on Fourier transform, basically, the other way, because this kernel can be viewed as a convolution. Okay. Maybe it's not the best explanation, but this is the logic of this. Do I... Am I answered to your question? Yes, yes, thank you. Okay. Okay, so we have like two minutes more. Maybe someone will answer a question, sorry, ask the question. If there's no other, I'll ask like a stupid question. So, like, is there a potential Vx value which you can add, but still the betancers or the structure still holds? Not up to my knowledge. I mean, because really, betancers realize on... Okay, the idea of betancers is really to somehow express your wave function as a certain strange combination of plane waves with some wave numbers which are not just momentum as in a free system, but they are something, these are rapidities. And if you insert an homogeneity, you break down these hypothesis under which betancers work. So, up to my knowledge, there are... I mean, there are exactly solvable models with traps, but not... The exact solution is not based on betancers, at least not in the standard sense. Then, okay, this is an argument of recent interest, okay. It's an argument of research. So, probably there are some papers where they address this issue, especially considering the small perturbation, small line homogeneities, like being close to the homogenous case. But, okay, I don't know, but not in the standard sense. This is the answer. Okay. All right. So, thank you, Stefano, for this very nice nice talk again. And you for the invitation.