 Okay, first I would like to thank the organizers for inviting me, it's a real great pleasure, it's exactly 20 years since actually I came to do post-doc with Boris and those two years in Princeton and NEC were really great years and affected all my life of course and I'm very happy that I had the chance actually to be with you, Boris. This is indeed a kind of continuation of the work that Ego told you about, I guess that it was kind of difficult to grasp all the details that Ego said because it took us actually a long time to understand everything but I hope that this is a fairly simple story because it is more of application of what this dipole moment of the vortex is doing in life. So it's a work that I did with Ego but it's based on the work of all three of us with my student of Ram Klein and I would like to start with the introduction. So the question is how can we see these effects of these dipole moments, there are several ways that we can think about it, we can try to realize one vortex and to see how it processes or to scatter phonons but it is kind of a difficult experiment and we wanted to look at some different experiments and to start with I would like to tell you something about superconductors which probably you all know better than me. So when you look at the dynamics of vortices in superconductors we know that they are all related to dissipation and entropy production and the first thing you can look at is to ask what's the friction force that is acted on a vortex and the first work is done by Bardin and Stefan which they showed essentially that if you look at a moving vortex then you can calculate the normal current that goes through the core of the vortex and then essentially this dissipation there is affected is this friction that acts on the vortex. Then Larkin and Ovchinikov came and they showed that a situation is more complicated when you go beyond the linear response because you can excite the quasi-particles in the core of the vortex and they can get heated and then you get linear response, non-linear response I'm sorry. And then non-linear behavior of the frictions as function of the applied current. So we try to push the current, the vortices with the current and you see that it is non-linear, it is not a simple linear relation. So they also saw for example this is kind of numerical, they also saw that there is kind of instability to filaments creating this system and this is kind of numerical work which showed that How does it go? We show that you start with abricus of lattice of these vortices and you start to move them and then you see that at some point you get rows of vortices instead of the abricus of lattice. So it's just to show an example for that. So it's clear that in superconductors this full structure of the core is important and it plays a role and it can have very important implications. But if you go to superfluids then apparently nothing of this happens and the reason is that until I think our work people believe that vortices in superfluids, in bosonic superfluids are essentially have featureless cores. So there is no core if you want, there is core but it has nothing to do and then what, there is no analogous scenario to what I was describing before in superfluids, in superconductors. So people believe that the mass of the vortex is very small essentially zero and then you can apply Kelvin's theorem and if you apply Kelvin's theorem then you cannot apply a force on a vortex because there is no mass and then vortex just moves with the flow, that's it. And there is nothing special. I mean the fact that the vortex moves with the flow doesn't mean that it is not interesting, it can be interesting that just curiosity, this is kind of a, I'm sorry. This is a, well it doesn't work. This is a two vortices and two anti vortices in just without any core, without any structure of the core and you can see that it has chaotic motion and it is because of the interaction between them and it is well known and so it can be also interesting but what we want to say is I believe another layer which makes things even more interesting. So this is the pictures Igor showed before of the core of the vortex and so we claim that you have to take weak solutions of the Grof Spitevsky equation in order to describe fully the systems in the weak coupling regime and this means that there is this cut near the core of the vortex which these are the pictures Igor showed. What this cut implies, I just, I want to summarize more or less what Igor said is that actually there is several energy levels, low energy levels, low, meaning that they are low with respect to the phonon band in this system. So there are several energy levels, these are the energy levels and that these energy levels are long lived because the time it takes for a vortex to emit phonons is very long. It goes like omega to the power fourth where omega is the, if you want the cyclotron frequency of this vortex. So the spectrum of these vortices schematically looks like that so you have several energy levels and they are essentially well defined. So this is the spectrum and what we want to understand is how this spectrum affects the behavior of vortices in the system. Now again I just want to say about the weak coupling regime, weak coupling regime means that this lambda which is the dimensionless coupling constant or interaction constant is one over the number of, if you want particles in a healing length, in a square of a healing length, which is, if you turn the interaction to be very small it can be arbitrarily large. And in principle you can realize this for example in cold atoms system. Okay, so what we want to say is the following. We know that direct observation of one vortex or two vortices is difficult but what we hope is that to look at the effect of this energy levels in the vortex on the collection of many vortices in the flow of them is more easy. And the idea is that we have many excited states which are relevant and long relaxation time which may generate heating. So we may heat the vortices and they get nonlinear behavior of the flow. Okay, so this is the idea, it is kind of similar to Larkin of Chinnikov scenario. So this is the outline of my talk. I will first show what is the experimental set ups which we think about. There are two kinds and then I will present just the results. Then I will present the effective Hamiltonian of the system and some qualitative discussion how you get the results. Then I will do some more quantitative calculations. I show just schematically, I mean just very fast how you write the kinetic equation, how you solve them and then I will summarize. So this is the artistic view of ego actually of how these systems look like. So there are two kind of set ups which we have in mind. One is by realization in BEC's called Atoms and one in Helium, a thin film of Helium 4. So in BEC what you can do, you can just cool the system and then you can put vortices by phasing printing. It is well known, people know to do it. Then you can, instead of moving the fluid through disorder, you can just move a disorder through the fluid and you just do it by sweeping, let's say, a speckle pattern through the system. So this is the system and what we want to say is the following, so you will sweep the disorder by velocity v. And this velocity v will affect the motion of the vortex itself, this is the vortex. So there are two components of the velocity, one in the direction of the flow which I wrote v perpendicular and one in the perpendicular flow which is v perpendicular. And then you can just record these velocities by measuring them. You repeat the experiment many, many times and you look at the velocities, these velocities as a fraction of the velocity of the disorder. A second realization which we think about is this Helium, thin Helium films on a substrate, what you can do is the following. You can put Helium 4 on some substrate, clean substrate, but our cleanities, there is always a lot of disorder inside and then you can rotate the whole substrate. What you do essentially is you rotate the whole cryostat together and it has been done by Bill Gleberzone so he told us that he did it and it is possible to do. And then you can induce some fluid by for example putting some evaporative heater at one side and some reservoir at the other side. So you can generate flow inside the system and again what you can measure, so you cannot measure some particular vortices but you can measure the difference of chemical potential or the pressure between the two sides. And he also invented, Bill Gleberzone invented some device to measure this and this difference of chemical or gradient of the chemical potential is proportional to the v perpendicular and you can look at this as function of v. So that's essentially the two setups which you can look at and these are what we expect to see the result that we calculated. So it's easier to present the result in terms of the drug coefficients. So drug coefficients is the proportionality factors between the velocity of the superfluid or the disorder in the case of the BECs and the velocities of the vortices themselves. So there are two drug coefficients and they behave in this form. So here this axis is just the velocity that I apply on the vortices and the different colors corresponds to different temperature of the phone and bath. So this is the low temperature and this is high temperature and here it is the same. So what you can see in v perpendicular here is that you start by essentially there is nothing happening which means that your vortices are pinned to the disorder. This is here and here and then there will be some maximum and then eventually it will go down. So when you start to take the vortices very fast eventually you don't have any, the effect becomes smaller. Okay so this is, these are the results and I just want to add more about if you just look at v perpendicular as function of the applied velocity of the superfluid. Superfluid then essentially it has some kind of n-shape similar to the case in superconductors which means that you have some, you have to reconstruct and there is filament instability in this system also in principle. Maybe there is other instabilities which occur before but in principle we know that there is filament instability here. Okay so I want to present the effective Hamiltonian so what we are going to consider is the following. We are going to consider a situation where the vortices are very dilute in the system. So we neglect interactions between the vortices and we assume that the disorder is sufficiently large so we can indeed do that. And so we can look at just the effect of one vortex. So the Hamiltonian which I'm presenting here is essentially for one vortex and what do we have here. So better is some constant which is not important I will work in dimensionless unit all over here. And so what is this log? This log has come from the core reconstruction. This is a mass which is non-analytic. P is the mechanical momentum and this is just a mass which is divergent in the limit where P goes to zero. So this is the first term. There is another term here, epsilon is just rotation matrix by 90 degrees. This term is eventually you can look it as giving the Magnus force. And these are the usual terms which come. This phi is the scalar potential from which you get the E. I'm working in proper formalism that Igor mentioned. So E is essentially the superfluid current rotated by 90 degrees. And these are the usual relations from electrodynamics. And B is the density of the superfluid here and it is given by a rotor of A, A is the vector potential. Okay, so and P is the mechanical momentum which is given by this term. Sigma is just the vorticity so it's plus minus one. So this is the Hamiltonian of the system and this order enters through this B. And to explain why, you can just write the Hamilton equation. So I just write the Hamilton equation for this Hamiltonian and I just combine the two equations together and I see that the P just depends on the gradient of B. So essentially we want to vary the superfluid density in the sample. We can do it indeed by several ways. As I mentioned then this will act as disorder in the system. Okay, so now I want to start the qualitative discussion and I need to define some time scales in the system. So I will write B as some average density plus some variation of the density. And this variation of the density is a random quantity, a random field. So I will characterize it by this correlator, the average is of course zero. And the correlator with Fourier transform with respect to the difference between R1 and R2 and the Q is the Fourier variable has the following form. So there is some dimensionless constant which characterize the strength of the disorder and notice this density. And G is dimensionless function which is a function of Q times Rc. Rc is the correlation length of the disorder. And G is some function we don't care so much about it. It's not so important for our considerations. What else I want to tell about this? Okay, so what we do assume is that this correlation length is larger than the healing length which is reasonable because the healing length is always pinning how much your density can change. So this correlation length should be larger than the healing length. And we shall work in the limit where if you want temperature is sufficiently large so that we start with P with the mechanical momentum which is higher than this one of the correlation length. Which means that in this system we have small angle scattering. Okay, so the disorder does not make large angle scattering but each time vortex is scattered is scattered only by small angle. And if you look at the transport relaxation time then just using Fermi-Goslin rules with these equations we will get the following. We will get that it is proportional to omega c. Omega c is the procession of the vortex, vortex processes all the time in our system. And then there is some scale of epsilon d of the disorder and scale of the kinetic energy of the vortex and it goes to the power 3 halves. It's just simply Fermi-Golden rule. So this is the transport time. Now there are two limits which we can look at. One is the limit where the kinetic energy is larger than this epsilon d, this one here. And then essentially what we have is that the vortex just changes many times, processes many times before it changes its position. In the opposite limit what happens is that it is changing its directions even before it completes one cycle. Now there is another characteristic time scale and it is the time scale in which you scatter over all angles no matter whether they are small or large. And it is this which we wrote TQ, it's the elastic time. And what we shall assume is that omega c time TQ is larger than 1, which means that essentially we can forget Schubnik of the Haas kind of interference effects. So this is the limits where we work at. It implies that this time, this energy scale should be larger than epsilon d and it turns out that indeed it works well. Okay, so now let's look at the vortex in a flow. So this is the flow, v is e divided by b, this is essentially the velocity of the flow. This is the vortex and let's just for convenience look at a frame that moves together with the vortex. So let's first consider the case where there is no disorder. So it's tau transport goes to infinity. So if there is no disorder we know that just the vortex would go with the flow. So in this coordinate system it will be fixed. Now what happens if we start to make the disorder a little bit larger? Then if we start to make the disorder a bit larger, this omega c is much smaller than 1. It's essentially omega c tau is the angle in which you scatter. And then when you start to add disorder you will start to generate velocities in the perpendicular direction. And you expect to have because of the flow something which will go perpendicular will be proportional to the small angle and to the flow which is e divided by b. So you will start to generate this. This is at small velocities. Now what happens about power production in this case? So of course the minute you generate velocity in the direction of the field e you start to generate entropy and the heat. So what is the power production? Just according to the usual equation it is the charge which is the velocity times the velocity times e in the same direction. So I just plug the velocity which I got before here and I see that it is e squared over b times epsilon divided by epsilon d to the power 3 half. So if I look at the power as function of e at the beginning it will grow as 3 to the power 3 half. Now what happens when I go to the large limit where omega c tau is much larger than 1? So this is the limit where the kinetic energy of the vortex is already very very large and it is less affected by the disorder. Because it is already high kinetic energy and then you expect that actually the effect of the disorder will become smaller and smaller. And on dimensional grounds what you expect is that in this limit you only rarely change your position, the vortex changes its position and then it will go like 1 over the omega c tau and omega c tau is very large. And then you expect to have actually this behavior that it goes like epsilon d divided by epsilon to the power 3 half so you go down again. So there is some kind of big structure. Okay so the power production is non-linear and it is quite reasonable because we indeed expect that at very small driving forces the effect will be small and when you get to very high energies the vortex is already not very affected by the disorder. And then it will be smaller. Okay so what does it mean? So if you start with the vortex at some energy then and let's at the beginning assume that there is no any dissipation by phonon emission then it will be excited to higher state and you will start to accumulate more and more energy. Of course it will stop by phonon emission and if there is some kind of time scale for inelastic processes this time scale tau in then you will accumulate this work for tau in and then what you expect to change the energy by tau in times the power that you accumulated. The power that we have time so this is the work and it has this form. It means that if you look at the distribution function as function of E then there will be a very long, a very large interval of energies where essentially the distribution function is fixed and this is the form of the distribution function. So at very small energies you will get just the thermal behavior, this exponentially decay for the thermal then you will have a very large shoulder because essentially you don't care what is your energy it will be always fixed. This is like effective temperature and then you will eventually decay again. Again thermal because the effect of the heating becomes very small. Now if you look at the non-equilibrium currents we know that non-equilibrium currents go like a proportion to dfd epsilon to the derivative of the distribution function with respect to epsilon and then you see that there are two peaks at the beginning and afterwards and this is the source of this non-linear behavior that I told you about. So this behavior when you just integrate the velocities with this kind of two peak function then you can explain this behavior of the drug coefficients which I told you about. You can understand that you first start with small and then you go to some peak and actually at a long time eventually you will be suppressed just because these peaks will all together go down. Okay, so this is more or less the qualitative explanation of the result. I know it's not very qualitative but this is our best. In order to calculate really things you have to write the kinetic equation and this is the kinetic equation. So what we assume we just suppress spatial dependence and we take only the angular dependence of the system. So I have, this is just the usual term. So if I forget the collision integrals and I just solve these two terms then I get essentially just procession, simple procession of the system. But there are collision integrals and there is one that is inelastic and one that is elastic. And the reason that I have elastic collision is just you can understand it in the following way. Imagine that you are in the moving frame of the vortex. So the vortex sees varying potential and it is like time dependent similar to time dependent force which excites the system to higher. And when you write this equation you get that it is just this form. The inelastic term has also this form. It is where T is the phonon temperature. You can see for example that if we don't have anything else and we solve this you get just the usual term and distribution function you expect to have. And maybe I should stress that this tau inelastic is very long time. It goes like 1 over tau goes like omega fourth and I remind you that the cyclotron frequency is very low. That's the whole point of what Igor said before. Okay, so we have to solve this distribution function and to calculate the velocities we just have to calculate this form. It's just d epsilon dp is just the velocities and the cosine phi or sine phi are just the directions. So this is how I choose. So this is the direction of v perpendicular which is the same direction of the flow and this is v perpendicular and this is v parallel. You just have to plug the distribution function here and to calculate it. Okay, so how do we solve it? So we solve it in perturbation theory. So we assume that there is one part F naught which does not depend on the angle and then there is another part which is integral over the angle is zero. And then what we solve in perturbation theory in this term, in this angular term. And in this approximation we can forget about the inelastic processes and just find the relation between F1 and the derivative of F naught which is independent of the angle. And this is the expression and then we take this expression back here and back to the equation and find what is the equation to F naught. And it turns out that F naught has this simple form. It is dFd epsilon equal F divided by something which is like the effective temperature of the system. And this effective temperature is the phonon temperature plus what I wrote before as delta. So this is the power production over time tau inelastic. This is essentially this term. And then we take this result and calculate V, the velocities. And these velocities again you can see that there is this term that depends on this double-picked function. And you can calculate V at large energy asymptotics and small velocity asymptotics and get this expression for the drug coefficients. So you match the asymptotics and in order to do that you have to find what we call the threshold velocity, which is the characteristic velocities of going from one solution to the other. And I write, we wrote these velocities in this form so that it will be clear that this velocity is much smaller than the speed velocity. So each term here you can check that it is smaller than one. And here again what we have is the superfluid density, the healing length, the vortex mass here, which is the mass Igor mentioned before, the temperature of the phonon bath. And here it is and everything else I already defined. So what we want, the reason we wrote it in this way is to stress the fact that we can get all these nonlinear effects well beyond the speed velocity. So it is relevant and it can be found. Okay, so let me summarize. So there are low energy excitations of the vortices and these low energy excitations are long lived and the vortex in principle can be heated. And if it can be heated then we have a nonlinear transport phenomena which is similar to what happens in superconductors. And another thing we want to say is that actually if there will be an experiment confirmation of this nonlinear, it will be a smoking gun for these nonlinear reconstructions. And that's why we think that it will be very interesting to see these effects. Thank you very much.