 Спасибо, Эмиль. Ну... Движение очень долго, но вы увидите, что все, что я говорю о сегодняшнем, это больше всего. Моя главная часть этого работа и самая créативная часть этого работа была сделана от Абрахам Клян Кибро-Университетка, студентом Адеда Гам, и мы опубликовали довольно долгое тему в 2014 году, и все, что я говорю, это в результате этого тему. Это то, что я буду делать сегодня. Во-первых, я попробую формулировать проблему во-вторых, и формулировать notion of the dipole moment of the moving vortex, and then formulate the result, which was quite unexpected, at least for us, which will be about non-analytic dependence of the dipole moment on the velocity of the vortex. Then I will have to spend some time describing the formalism, the motion of the quantum vortex is in the superfluid, and this is known as Popov's hydrodynamics, and I will remind to you what it is and how to work with it. Then I will switch to our results, which is the solution of the moving vortex in Popov's formalism, and there will be first critical solution, which is due to Avraham Klein, then there will be some analytic interpretation and the connection to the so-called weak solution of the Gross-Pittayevsky equations, and then I will derive the answer for the dipole moment of the vortex, which I will formulate in the first part, and then I will discuss the physical quantities, клатрон dynamics of the quantum vortex, and we will talk how it comes about, why those excitations are long-lived, which brings me to the point of the interaction of the vortex with phonons, which will be the problem of scatterings of the phonons and the radiation, and then I will conclude with the quantization of the vortex motion, and will show the new low energy degrees of freedom, which will appear after the physics. I will discuss over here. Basically, I will start with a very simple problem, which we start teaching our students about Sutter-fluid hydrodynamics. I will talk about the quantum vortex in local hydrodynamics, and basically what you do, you introduce one field is the velocity, and the other field is the particle density. Then the first equation, which you write, has nothing to do with the Sutter-fluidity in the stationary case. It's just a continuity equation. The second equation is already quantum. It replaces the Kelvin theorem in the classical hydrodynamics, and basically the statement of the vorticity is quantized, and the statement goes back to Anzager in 1949. So this equation is classical, this is the equation, which is quantum. As a matter of fact, you can see that you can quite easily get rid of all the constants, just change, just measure r in units of the healing length, which I will introduce later on, but for starters it just has a meaning of the size and velocity and then has a natural units h bar over mxi or one can quite easily realize that it's nothing but the sound velocity in the super fluid, and if you do this, then those equations become dimensionless and you have to close the system and you are closing the system due to Bernoulli equation, and once again there is nothing quantum but equation and in the proper units it just has the simple form. So then you can quite easily solve the first equation, it has an obvious form in the complex notation like so, then you put it into the Bernoulli equation, it's at the large distances it's one, then of course it's down near the vortex as the algebraic function because this we square it over two and of course this formula is correct only if z is larger than one and then the local grid dynamism become unapplicable and you have to solve something else which we will talk about later on but z equals to one you can basically find the definition of the healing length from here this is the distance which drops by a factor of two so and then you have to check the last equation which is a continuity equation and you quite easily see that it goes through just you see that the real part is here you differentiate with respect to z the derivative is coming only from here so the continuity equation goes through and you are done with the vortex and you are done with the motionless vortex in terms of the local grid dynamics so everything is consistent the problem begin if you start considering a little bit more complicated problem which is the following you just fix the vortex you just fix the vortex at the position equals to zero and put the boundary conditions that you have a superfluid velocity at the distance going to infinity so by the virtue of the Galilean invariance you can ask the problem of the vortex which is moving with the velocity minus with respect to superfluid very well let's try to do the same thing you have the same equation but the but the initial condition changes but in principle you can try superposition so that what I do once again you have to go through the Bernoulli equation it doesn't change so it's like this you have to put this velocity take the modulus squared and then you try to go and then you try to go through the continuity equation and you see that due to the fact that here you have v squared and you have a product of constant in this star then as a matter of fact the continuity equation doesn't work due to the fact that the liquid the liquid is compressible if you are non compressible it would work very well but here you already have a problem so and you can try to resolve this problem using the iteration which is quite easy thing to do and you get the first correction which look like this and as you see see this combination has the meaning of the additional flow which is created by the dipole which I will denote by the letter D and you can find the expression for this dipole moment and the first term is proportion to the logarithm of Z and this is the deformation this deformation is the distances larger than the healing length but it also contains some unknown constant which I will call the dipole moment on the healing length now looking at this formula basically this is not that it's something extremely new people knew this this for many years the point was always that this comes from the wortascore deformation therefore it cannot contain any large factors therefore whatever calculation of the wortascore deformation you are doing it's basically just change the factor in front of the logarithm there is practically nothing to study the point which I am going to make that it's not really so and the result is yeah and I forgot to mention that why this dipole moment is important it's important that if you start thinking about the wortasc's mass you immediately realize that if you think about the wortasc's as a point particle and then think about the superfluid as something which you move together with the wortasc's you know from the course of hydrodynamics that the added mass to the body is proportional to the dipole moment of the superfluid so as a matter of fact the problem about that dipole moment is the problem about the effective mass of the wortasc's so anyway but the point of our paper and the point of the point of what I am going to talk about during the remaining time that this dipole moment as a matter of fact is not the simple analytic function which determines the height of the logarithm as a matter of fact it's a function by itself which extremely nonanalytic in terms of the velocity so as a matter of fact what I am going to talk about that the deformation of the wortasc's core as a matter of fact doesn't have a Taylor expansion а So а Now Before I start presenting the detail the detail theory and the numerics I have to I have to talk to you which equation we actually use in order to describe the wortasc's system Because the gross-petayevsky equation is not extremely suitable I will explain later one and we used what is known as Popov's grid dynamics for two dimensional both of gas this is in the textbook in chapter 621 So and basically the story is the following. You start with the gross-petayevsky action and in two dimension it's just the standard bosonic action and the energy has a form Если вы понимаете, что Лиандра — это демонсиональная интеллектурная стресса, то вы понимаете, что в том числе эффекта, если вы меняете дистанцию в том числе хиллинга, в том числе времени, в том числе инверсии хиллинга и энергии, в том числе сайта квадрата, в том числе нот, и просто redefine that the healing length times the density is one of the lambda, which is basically the large parameter of the theory. In theory, you realize that it's a matter of fact. All the action doesn't contain any constants anymore, and lambda plays a role of the effective plant constant, so that if the interaction strength is small, then the semi-classics for this action is perfectly justified. Now, the problem is that at the large distance this term is much more important than this one. The effort to work with this term in terms of the perturbation theory is not convenient, and it's not practical. Once again, what you do, what is known as a long transformation, it's all standard, and what you obtained from here is the expression in terms of the density and three-dimensional velocities, and each component of the velocity is determined by either time derivative of the face or the coordinate derivative of the face, and it's all quite standard. The problem is that, as everybody knows, that theta is a multi-valued function of x, which means that any jumps by any integer factor of 2pi on any semi-plane is allowed. Therefore, if you would like to consider the vertices, you will have to introduce the lines, then you have to introduce in 2 plus 1 space as a semi-plane, which goes to infinity and allow jumps by the factor of 2pi, and if you would like to consider the vertices, you have to be able to consider the moving of the boundary as a function of space. And this is very complete, and this is quite a cumbersome thing to do, and the idea of pop-off is basically the following, that you trade the velocity field to some auxiliary field using the following recipe. You basically say that I have three components of the velocities, and just let treats, and if they were independent, it would be the Gaussian series, so, in principle, you can integrate over V. However, as a matter of fact, the three-dimensional curl of the velocity in the absence of the vertices is zero, because it's a three-dimensional gradient of the function of theta, and in the place where you have a vertex line, it determines by the vertex density current, which I define like so. This line, this line, this is the expression for the simple vertex density, which basically shows that it's just a delta function, which moves according to this function r of t, and this is the vertex velocity, which is nothing but the vertex density, which is multiplied by the vertex velocity. And, of course, by construction, the vertex current is conserved, and it means that it means that similar to the motion of the particles in the classical electrodynamics, you can introduce the gauge theory. And this basically was done by pop-off, so, what you are doing, you put this constraint as an integration of the auxiliary gauge field with three components. This is scalar potential, those two components of the vector potential. So, you have a function, which is linear in V0, and the quadratic in V, so, you can calculate the functional integral, and this is a pop-off action. So, basically, it contains two parts. First part is precisely what you have in classical electrodynamics. This is the vertex current, which is equivalent to the particle current, multiplied by the vector scalar potential A. And even though here you have explicitly the vector potential, the theory is still gauge invariant, because the vorticity is conserved. And the large part, and this part basically includes all the phonons. It includes Bogalubov quasi-particles, it creates phonon-phonon interaction, it contains everything you can think of. And you see that if you put B equals to 1 over here, you forget about this term, and you just count B from 1, then you restore E squared minus B squared over 2, which would be linear, which would be linear electrodynamics. Now, here it's nonlinear, it's nonlinear electrodynamics, and it's nonlinear electrodynamics for the reason that the symmetry of A squared over 2B is a combination which is protected by nothing but the Galilean invariance, which I will show later on. Now, if you would like to go back and try to establish the dictionary between old variables and new variables, you see that the effective magnetic field is nothing but the density of the particles, and the current is nothing but the electric field rotated by the 90 degrees. So, okay, this is more or less Galilean invariance, and as I said, the form of this term is protected, and here you can write a principle whatever you want, because under Galilean invariance B doesn't change. So, we have an action, we have an action, and therefore you can write an equation of motion, which will be equivalent to the Maxwell equation. The first Maxwell equation follows from the connection between the gauge fields and the physical fields E and B, and if you think about meaning of B and meaning of the electric field, this is nothing but the continuity equation, which is written in a new variable. Now, the second Maxwell equation, which used to be divergence of E equals to density, becomes divergence of E over B equals to the vertex density, and this what replace the quantization of the vorticity, and it's once again much easier to think about the original equation, and this what Maxwell equation looks a little bit terrible, however, if you forget about those nonlinear terms, this becomes just a regular Maxwell equation, right? Now, this term, once again, is needed by the Galilean. This term is once again needed for the Galilean invariance, and this term is something which defines the physics in the world of score. And once again, this term and this term together, they are fixed and the necessity of this combination is from the vorticity conservation. Now, as a matter of fact, Popov equations allow you to do, to extract something more, which in the Gross-Petaevsky is kind of obscure, namely to immediately calculate the force acting on vortex, and this is quite easy, you just vary this term with respect to the position of the vortex, and you find that force actually consists of electric of the force due to the electric field, and the Lorentz force, and as a matter of fact, it's nothing but the Magnus force, and moreover, you can check that this is the Galilean invariant quantity, as it should be. So, Popov's equations are nonlinear, and as a matter of fact, the disentering of self-interaction is non-trivial, and I will talk about this issue later on. Okay. Finally, to complete the picture of the Popov description, let me comment about the relation of the Popov's equation and Gross-Petaevsky equation. This is the Gross-Petaevsky equation, which is nonlinear Schrodinger equation. Though the Popov equations, you can basically just check that if you put b equals to psi squared and e equals to the current rotated by 90 degrees, then those quantities go through, provided that the coordinates is not coinciding with the position of the vortex, and psi squared not equals to zero. Otherwise, in the region where psi squared equals to zero, the Gross-Petaevsky and Popov's equations are different, and we will see later that as a matter of fact, it's important. So, now, this is about Popov's story, which was done quite a while ago, and he didn't get too much out of this. So, what I'm going to do now is to basically come back to the story. I told you in the very beginning about the dipole moment, and to talk about how this dipole moment appears within Popov's formalism. So, the first question is, of course, just isolated vortex to make sure that everything is correct as it should be. So, you basically take the second Maxwell equation. Solution is quite obvious. Then you put it back into the third Maxwell equation, and you see that B, which you are getting from this equation, is rotationally invariant. And then you check that as a matter of fact, it immediately goes through the first Maxwell equations, that E, indeed, identical equals to zero. So, this equation is the one which you have to solve, and you can change the variables, just to make sure that you obtain the result, which was obtained by Gisburg-Petyevsky in 1958, and the only thing which I will need from those equations is that at the large distances, at the small distances, the density goes down as a second power, the coordinate with some constant, which will play the role. And what I am going to construct further. Okay, then let us do a little bit more complicated thing, then let us fix the vortex. And in Popov's formalism, you can do it quite easily, unlike in Gisburg-Petyevsky. Once again, put the condition of the finite superfluid flow at large distances. So, this is a curl of E equals to zero, so you can parameterize it by the scale of potential and get a set of the coupled equations. And those set of the coupled equations were solved numerically by Abraham Klein in 2012-2013, and also something was done by Vadim Chlanov. And the result, the numerical result, which I am going to present you first, is basically the color plot of the density, the color plot of the density as a function of the coordinate, so superfluid velocity looks in this direction, so the Magnus-Force director perpendicular, this should be well. And the distribution of the density. In the first glance, you can see nothing, right? Just density vanishes somewhere, and it's a little bit not cylindrical symmetric, which indication of the dipole point otherwise nothing spectacular. However, if you blow up a little bit, you see that something is already going on, and if you actually try to measure the density, as a matter of fact, you will immediately figure out that immediately after the velocity equals to zero, the region where density equals to zero is not a point anymore, as it used to be for the motionless vortex, that as a matter of fact, you immediately have the line where density of the particle equals to zero identically. So, if you wish, what is the conclusion is that immediately after you start moving vortex with respect to the superfluid with the finite velocity, you get line of zeros, which also can be swapped as a sequence of the vortex anti-vortex pairs, which are thus close together, similar to the Swinger mechanism. So, this is just a little bit more information. This is just a density as a function of the coordinate along this direction, and this is a density as a function of the coordinate along that direction. Red dots are the numerical data, the black line is a theoretical formula, which I will show you later on. Okay, this is the distribution of currents. You basically see that there is a region, there is a line of zero density, and the current, of course, doesn't go through this line now. And the final piece of the numerical data, which you should know, that if you just find the numerical, if you just find the numerical, the length of the cut as a function of the superfluid velocity, you can see that more or less it follows the straight line. What? If you have a question, ask me. I know better, I think. So, basically after I presented numerical data, I will try to explain why do we need the cut as a matter of fact, as a matter of principle, and why is the moving vortex, as a matter of fact, is not compatible to the strong solution of the Gross-Petyevsky equation. And then the second question, which I will try to answer is how to describe the solution near the cut and how to reconcile the analytics with numerics. So, let's suppose that I have a Gross-Petyevsky equation, and far from the core, the solution for the moving vortex, once again, is just a phase, which has to satisfy superposition principle. This is the phase of the vortex and this is the phase, which corresponds to the superfluid velocity. At the distances, at some intermediate distances, you can expand it. Right? And then, basically what it looks like, what it looks like is just a Bogaliubov expansion of the Gross-Petyevsky non-linear equation, which I will write like this. And then, though the just stationary Bogaliubov equations, which you are supposed to solve with the boundary condition, that functions u and v should look like this. So, and this Bogaliubov equations, as a matter of fact, you can solve quite easily. The first equation, the first solution is kind of obvious, and it works independently on the density in the vortex, and this is just a translational invariance, that if you shift the vortex, it will still be the solution, right? So, and this is what you do, and this is the exact solution of the Gross-Petyevsky equation. The second equation you can also, the second solution you can find quite easily, because there is a Vronskin, which equals to constant, and there are, and of course, the solution has nothing to do with the solution we are looking for, because it's a symptotic behavior, it's one over r, and we are looking for r. Now, here is the second solution, and you see that it has a correct asymptotics, it's the large distances so far. However, if you go with the distances smaller than the healing length, what you find immediately, which means that if you would like to stick with the solution of the Gross-Petyevsky equation in all area of the space, you immediately figure out that you have to throw away the divergent solution, which means that the solution of the Gross-Petyevsky equations, responding to the vortex, which means that the solution of the Gross-Petyevsky equations responding to the vortex moving with respect to the superfluid simply doesn't exist. And I believe this conclusion was first noticed by Tauly Centenglia. Now, then, as a matter of fact, what we saw numerically is that you have to allow for so-called weak solution of the Gross-Petyevsky equation. What it means is basically the following, that you allow Gross-Petyevsky equations to be violated in the set of measure zero, which is a straight line, which in this particular case is a straight line, and you can figure out immediately that the action is finite, so action doesn't diverge. So this equation realizes, as a matter of fact, the minimum of the Gross-Petyevsky action in some extended sense, that you are allowing for the weak solution. And if you do so, then you can solve all the problems quite easily, because in the vicinity of the core Gross-Petyevsky equation is nothing but the Laplace equation, you introduce the cut and put the condition density on the cut equals to zero. Well, if you have a cut, you can do Zhukovsky transformation, of course, and find the solution and basically make a conformal transformation of the cut onto the strip. Do and look for the analytic function satisfying by the boundary condition and the only function, which you can possibly write here and you cannot write anything else. And you look at the asymptotic behavior and you immediately find the expression for the width of the cut in terms of the superfluid velocity and you see that the length of the cut is proportional to the superfluid velocity with constant alpha squared, there is nothing but the constant in the constant in the Gisburg-Petyevsky solution for the isolated vortex at vc equals to zero. Now, it requires some work to calculate the phases and to figure out that as a matter of fact that vortex is located here, meaning that if you take this integral around this contour, you will take the proper 2 pi and if you take this integral of any combination around this contour, you will get zero. It's quite easy to do, but I will not have time to explain it in details. Now, once again, this is numeric versus analytic. This is length of the cut multiplied by this value alpha squared, which has to be the straight line, which is shown by the black line. Those dots are numerics. This black line is an analytic formula and you can see that whatever numerics we can do, we are within the limit now. This is the numerical data for the density, those are analytic formulas, which I showed you before and once again you can see that it is as good as you can get. Okay. Now. So, anomalous dipole moment. You have a asymptotic behavior corresponding to the weak solution. You have a bogale of expansion. You can match this one with this one in any distances, which you like. You can get basically the dipole moment expansion, which I promised you in the very beginning, which corresponds to the local grid dynamics and the expression for the dipole moment follows from here and this is the formula, which I showed to you in the very beginning. So, the first intermediate conclusion is that for the superfluid moving with respect to the motionless vortex, there appears the anomalous dipole moment, which doesn't have a Taylor expansion in terms of the velocity of years and this is not a consequence of the large distances. It's a consequence of the distances smaller than the healing length and it corresponds to the non-analytic reconstruction of the vortex core, which corresponds to the weak solution of the Gross-Petyevsky equation. Now, strong solution of the Gross-Petyevsky equation as I said, doesn't exist. I will prove this formula a little bit by immediately noticing that you can play the Galilean invariance. The dipole moment has to be Galilean invariance. So, instead of the superfluid velocity, what you have is just nothing but the force, which acts on the vortex, which is the total Magnus force. Now, I explain to you how this thing appear. The last part of the talk and I believe that Agam will continue is why should we care. And, as a matter of fact, we should. So, what I'm going to do is to start to derive the effective low energy theory, meaning that I would like to consider distances much larger than the healing length and the frequencies much smaller than the healing energy and basically what I will try to do is to take full flash pop-off action and introduce it to the low energy action. Now, if there are no vertices, this is pop-off and, of course, if you go with the large distances, what you can drop immediately is gradient terms and you end up with the action which is still nonlinear and it's nonlinear for the reason because the Galilean invariance which distances you are. Now, let's consider dilute vertices in the first in the original theory in the original theory there were the points which were moving with the same velocity. Of course, you would like to keep them as a point, but you cannot because you got rid of those derivatives over here which means that if you keep them as a point, the solution will be badly divergent. So, as a matter of fact, what you have to do is to regularize the vertices current. There are many ways to do it and the way we found very practical is to consider instead of points, the ring with a conserved vorticity and the reason why it's inside the vortex equals to zero. So, as a matter of fact, what you feel is only the fields which are created by all the other vortices and therefore it's a way to automatically take care of the self-interaction and of course you have to take into account all the non-analyticity and reconstruction I talked about and this is durable and here is the result basically this is the action, this is the vortex current which is regularized by the transmitting delta function into the circle and the last term is the contribution from the core of each vortex. The first term is just the manager of the core which is the function of the external bazonic density and this is known the second part the first part was not known before and this is just a contribution of the dipole moment I was talking about that if you differentiate this one with respect to F you will get the anomalous dipole moment I was talking about previously. So, this is the effective action you can derive equation of motion the first equation just a continuity equation for the particles is the same now the first Maxwell equation you have a density you have a density vortex but you also have a contribution for the dipole moment of the vortex similar to what you have in electric dynamics of the continuous medium now if you have a contribution due to polarization of the vortex to the density you have to have the contribution to the vortex density due to to the current, due to polarization and once again they are connected by the conservation of the vorticity and you have and you have a contribution to the magnetic moment of the vortex due to the fact and you have a contribution to the vortex current due to the fact that the energy of the vortex depends on the effective magnetic field now there are two terms one of them is due to the current in the core and the other one is once again due to the Galilean invariance due to that that if you move that if you move dipole with a finite velocity once again, due to the Galilean invariance you have to have a magnetic moment so and of course there is an equation of motion due to the vortex and you will see that now in addition to what is done before the kinetic energy appears this is once again Magnus force and this is force on the dipoles on the electric and magnetic dipoles in the homogeneous fields right well and the smallest of the equations you can and consider the motion of the vortex in the frozen field what you get what you get what you get are the oscillations and the period of the oscillations of the oscillations is the function of the amplitude of the oscillations and what is remarkable that due to the non-analytic structure structure with the decreasing of the amplitude you can make logarithm as large as possible so that the frequency of the oscillations in the classical fields as a matter of fact with the increasing with the decreasing of the amplitude going down and it goes below the boundary and it goes below the boundary of the phonon spectrum if so if so then you have to you can analyze the lifetime of those oscillations with respect to the radiations of phonons and it's quite easy to do the remarkable thing that what you are getting is omega to the force which means that as a matter of fact those oscillations are pretty much long-lived oscillations and it can be it can be considered in the spectrum SN excitations this I will not be able to explain this I will skip and let me just finish with the quantization of the vortex motion as I said the result for the lifetime of the oscillations of the oscillations of the vortex with respect to the radiations of phonons is parametrically large which means that the quantization of the vortex motion is pretty much legitimate procedure what we did is the semi-classical quantization of the section of the effective action of the vortex and what you obtained is the following conclusion for the discrete levels now the pre-factor is nothing but the boundary of the spectrum now point is that a node size squared for the systems with the large scaling length is a large parameter which means that for some numbers of J J those levels are once again within the vortex the spectrum of the vortex therefore they have to be considered pretty much accurately together the spectrum provided that node size squared is a large parameter so those are my conclusions I basically try to convince you both by analytics and numerical argument that you if you solve the structure of the vortex core in the both interdimensional bos system with the large scaling length you will get anomalous dipole moment on analytical reconstruction of the vortex the leads of the dynamics of the vortex motion much slower than it was previously thought and therefore there is something there are some things to do and one of things will be the effect of those degrees of freedom on the vortex motion disorder potential which I believe will be considered in the next talk