 Okay, first of all, I would like to thank the organizer for inviting me to this brilliant conference. And this is, oh, where is it? Does it work? No? How does it work? This one? Where is it? Ah, I see. So what I'm going to speak about is this Bozanstein condensation. I don't almost see of the magnets in Ethereum Iron Garnet. And this is work which I made together with my former student, Pushan Lee, now post-doctor at Los Alamos, and another student, Chen Sun, and my friends and coworkers, Dr. Swine Seslow from Texas and the University, and Dr. Thomas Natterman from Cologne, from University of Cologne. And my sounds are to Sergei Democrito, who was the main person in the experiment here. Okay. Oh, sorry, I need previous one. So that is my congratulations to Boris. And on this, on this photo, you see that, on this picture, you see that Boris exerts some attracting force to other people. They lean to him. I'm one of multitude of people who experience this attraction, and I'm happy that our world lines crossed at some point, and I have a privilege and pleasure of communicating with him. Thank you, and many happy returns of your birthday. So what is, this is brief outline what I'm going to speak about. First, I will give a very brief introduction what is Ethereum Iron Garnet, and what is spectrum of magnets in it. Then I will describe the experiment produced on condensation of magnets in Ethereum Iron Garnet, because without that, it will be non-understandable what will be later. The third point is a theory for this experiment, and then I will speak about a new not yet published material, superflated of magnets. Only theory, no experiments so far. Experiment is in progress now. And yeah, conclusions probably, I hope that there will be time to read them. Okay, this is Ethereum Iron Garnet, it is very, what we need to know about it is that it is, do you see something? No, I don't, ah, oh, this is a very, very good crystal. Very good insulator, and therefore, spin waves in it are surprisingly, have surprisingly small line beads, and this is a best toy for experimenters in magnetism. They produce spin waves in it and transfer and made some interesting things and applications to that. I'm sorry, but I do not have time to speak about technical applications of that. Now, it is sufficiently complicated, it has sufficiently complicated unit cell with 80 atoms in it, and 20 of them are irons, there is ferrite with sufficiently high critical temperature and with big spin per unit cell, it's about 15. Now, the experiments on condensation were produced not in the bulk crystal, but in the film, and the film was not very seen it was about one to five microns, six, so it looks like it is three-dimensional situation, but in reality, it is not completely so because just at this sickness, the dipolar forces play the same role as exchange forces, and therefore, especially at very low energy. And therefore, the dipolar force should be taken into account very seriously. So the Hamiltonian includes not only exchange interaction, but also dipolar force and Zeeman interaction. Because S is big, Holstein-Prima-Kov transformation works well and all the calculations can be made explicitly, at least for the lowest energy band in which the spin of a cell rotates as a hole, okay? Therefore, this calculation for produced many years ago already, and the spectrum was calculated. The initial quadratic Hamiltonian contains not only a standard term in which number of magnets or number of spin flips is conserved, but also non-conservant numbers of spin flips here, so non-conservant magnetization. And that is due, these two terms are due to dipolar interaction, surely. Okay, all this, what enters in Hamiltonian and what enters the spectrum of magnets is written here. And as a result, we have, in contrast to the bulk spectrum, we have a spectrum which has two minima at finite wave vector q, q at minus q and maximum in the center, maximum at q equal to zero. This q is determined by dipolar interaction, also by exchange interaction. D is something like exchange constant multiplied by latest constant, by volume of elementary cell. And this is a sickness of the film, so this is sufficiently complicated combination, but in total, it has a value in the range of 10 to five inverse centimeters. Now, an important remark is that APA is an operator initially associated with the change of spin by one. Therefore, the operators of magnets which are linear combination of operators A and A dagger, in general, do not have definite spin, but fortunately, just near the minima where condensate is formed, u is very close to one and v is very close to zero. Therefore, condensate magnets have definite spin equal to one. Now, next approximation in the old stone expansion gives us the interaction of magnets and there are three different types of important interaction. One is the vertex with three lines which corresponds, in principle, to a decay of one magnet in two magnets, which is, which may be forbidden if the frequency is less than the doubled Z-man gap in spectrum. Now, next is the scattering process, two in two, they do not conserve energy and momentum of each, but they conserve a number of spin rays of magnets. And finally, one to three, these two, these and these interactions are due to dipolar interaction which does not conserve the number of magnets and this is forbidden if omega bigger than three delta. Nevertheless, it is very important for condensate because it contributes very unusual term to the self-consistent field. Now, I will describe the experiment. In 2006, the German group of experimenters in Münster under the supervision of Sergei Democrito discovered the phenomenon of both the condensation of magnets in Ethereum iron garand under the pumping. And what is also very important that this both the condensation happened at room temperature. So what is the basic experiment? They had a film of Ethereum iron garnet. The magnetization and magnetic field were directed parallel and they pumped magnets by a micro strip resonator under the film directly to Ethereum iron garnet. Because the electromagnetic field has momentum practically equal to zero, the only process which was possible or the most probable process which is possible was the decay of photon of electromagnetic field into two magnets with opposite momenta and with equal energy. Therefore, the energy of magnets was twice less than energy of photon. Now, the magnetic field was chosen in such a way. It is not unique way but you should satisfy this equation, this condition that this appear in magnets should have frequency smaller than doubled Zeeman gap, okay? Then the decay is forbidden and the number of these magnets is approximately conserved. Now, what they did after that, they considered, they observed how these appear in magnets, relax and form some equilibrium with condensate in the end. For that, they applied the brilliant light scattering. They had a laser focused in some point of film and the same lens collected the scattered light into some interferometer or filter which analyzed distribution of this distribution of these magnets by frequency. So that is what they observed in the left upper picture. Again, it disappears. In the left upper picture, you see the equilibrium distribution before they applied pumping. It is cut off because their filter had a limited band so it could not analyze further. Now, what happens after pumping? You see that the peak becomes more and more narrow and in the end, after 500 nanoseconds, the peak was so narrow that they could not resolve it. Okay, this is the same process but with angular analysis. And you see here that initially, they had a spot of appearing magnets. Then it becomes broadened in some range. The broadened increased, but after some time, it decreased and at 700 nanoseconds, it become very narrow near minimum. This point corresponds to minimum of energy in momentum space. So what is going on? I will present some simple theoretical ideas. And the first, they are based on approximate conservation of spin waves. Partly it was done by Bunkov and Volovykh. 20 years ago, they considered such a phenomenon in Helium-3. So the first important statement is that the pumping establishes a stationary number of spin waves of magnets, pump magnets, which is determined by the pumping power and the magnet's lifetime. So this is the pumped energy and it is divided by energy of magnet. Of pumped magnet, it gives the stationary number of. Now, then low energy spin waves relax to a metastable thermal equilibrium, but because they are conserved, this equilibrium will be with a non-zero chemical potential. Before of that, the equilibrium was with zero chemical potential because high energy magnets do not conserve their number. So they were able to confirm, experiences were able to confirm that what they observe is Bosenstein distribution, which can be substituted by Rayleigh-Jones distribution because the interest in energy are very small in comparison to temperature. Temperature 300 Kelvin at energy of Magnum, condensate Magnum is less than one Kelvin, approximately 0.1. So if you would have this distribution everywhere, the total number of magnets would depend on temperature and chemical potential. Initial distribution is the same but with chemical potential equal to zero and the difference between number of magnets with chemical potential and without chemical potential is just this fixed number of pumped magnets and you see here that it depends on chemical potential and this sum is converging even in three dimensions. Therefore, all these magnets are concentrated in low energy pumped magnets concentrated in low energy regions. Now, this is equation for chemical potential as function of number of pumped magnets. You see here that the chemical potential grows like as function of number of pumped magnets but it cannot overcome the gap in the spectrum because like in the standard was a conversation but in this case, the gap was zero so chemical potential could not be more than zero but here it cannot be more than gap in the spectrum and when it reaches the gap in the spectrum, this is just the signature of both a condensation. So the critical value of number of pumped magnets is just the difference between... So this value with mu substituted by delta. If the number of pumped magnets, if you increase the power of pumping and number of pumped magnets exceeds the critical value then the difference between the number of pumped magnets and critical number of pumped magnets fall into the condensate, okay? So you see here that the total number of particles in condensate also is completely defined by pumping power. Now, it explains why condensation is possible at room temperature. This is because the condensed particles all have the energy of the order of delta and delta is very small. Okay, six year later, the same experimental team made another very important experiment namely they improved the lens so that the spot of laser beam became less than wavelength of condensate magnets. And after that, scanning the laser, they could find how the density of condensate magnets depends on coordinate. This is this lower picture and you see very well seen interference structure. This interference is just the interference from two different condensate in the point Q and point minus Q. What was surprising in their experiment is that the contrast of this interference picture was very low, namely 3%. And I will explain next why it is so surprising. What brilliant scatter and see is it is local delta MZ associated with magnets. And if we consider only condensate then delta MZ associated with condensate contains of consists of is a square of modulus of wave function of condensate which consists of two superposition of two term from condensate in point Q and point minus Q. The square of CQ is a number of magnets in the point Q and this number of magnets in point minus Q and there is interference term with wave vector to Q. The fact that it is the wave vector is really to Q was confirmed by experiment very well. But if we would have NQ equal to N minus Q the contrast would be 100%. So that means minimum is zero maximum is something. But they had instead 3%. It can happen if one of two condensate is much smaller than another. If N minus Q for example is much less than NQ then this term is much smaller than that and we have almost constant. In particular when N minus Q is equal to zero there is no interference at all. So the question is why the symmetry Q to minus Q is violated? That is the first problem which should be solved. And that was solved in the work which we have published in the year 2013. And the idea was the following that the distribution of particles between two condensate is only due to interaction because the energy in these two points is the same. So redistribution without interaction would not change energy. So we can calculate the force order constant energy interaction between magnets and there are two constant characterizing this interaction. The constant A is interaction between magnets in one condensate, two magnets in one condensate. The B is interaction between two magnets in different condensate. And we have found that A is negative, B is positive for their conditions. If so that means that magnets in the same condensate attract each other, magnets in different condensate repulse each other. Therefore the ground state should be all magnets in one condensate, another is empty. But then if it is so there is no interference in the picture. So why interference appears? Okay, so let me write the most, it's almost most general equation for terms of force order. A and B are what we had before but there is also due to dipolar interaction. There is a term which contains one creation operator and three annihilation operators. You can take the wave vectors in such a way that some of all these wave vectors equal to zero. Take an account that you should, okay. What is the importance of this dipolar induced interaction is that it depends on the phase of the condensate. If you write that condensate is square root of number of particles times exponent of i phi, then the first two terms do not depend on this phi. But the third depends and in the ends we get cosine of some phase and this phase is some of these two phi. In the phi in the point q plus phi in the point minus q. Now you see that the minimum of energy definitely appears in the point when cosine phi is equal to plus one or minus one, okay. Cosine phi if c is a real number but you can keep it real. So that means that in principle first of all we have a phase coherence trapped phase between these two condensate. And secondly, what is also very important that there are two different states, pi state and zero state in principle. Okay, now imagine that we already made this minimization over phi and then what remains is the function of number of particles and q and n minus q. We should minimize this function under constraint that the total number of particles in condensate is constant. So what variable should be varied is only difference between number of particles in condensate and q and minus q. That can be made very easily because it is very simple. And the result is that the ground state depends on the criterion which is i minus b plus c. And if it is positive then the symmetric phase wins in which n q is equal to n minus q. If it is negative then non-symmetric phase wins. And delta is not equal to zero in this case. Now after that we calculated the phase diagram in the two variables. One is magnetic field, another one is magnetic field over, okay, this is magnetic field, this is sickness. Here are only small sicknesses but in big sickness practical only non-symmetric phase with phi equal to zero wins. So that is what now is available for experiment. But if we will have a sooner field in the range of between 10 and 20 then we will have first of all non-symmetric phase but with phi equal to pi. And secondly, symmetric phase with phi equal to pi. How much time do I have? Okay, not too much. Now I will show, so now after that I will show, I will should write the magnetization also in this form taken in account phases. And these phases also enter the interference pattern but not some, but difference. And this difference is a free parameter which simply shifts the interference picture as a whole. Therefore definitely doesn't change. The energy it is Goldstone variable. Okay, this is contrast, please note that it depends on model C and in the experiments A, B and C were like that. That gives for, according to this formula for beta, it gives beta equal from 2.5 to 5% and experiment it was from 3 to 10. And the theoretical reason for beta to be small is because QD is big parameter. In their experiment it was even not 30 but about 40. Okay, let me miss all that and that. Sorry, but I do not have time for that. But your slides are available from the program so people can see. Your slides are uploaded so people can see. This is Goldstone mode. We calculated it, we have sound like dispersion which becomes quadratic, a big Q. And interesting that this velocity of sound is about kilometer per second for non-symmetric case. But for symmetric case it is about 10 times less. Now about superfluidity of magnum gas. There are two different obstacles to superfluidity. The gas is, the magnum gas, the condensate is coherent. Therefore seems reasonable to expect superfluidity but there are two obstacles. First of all that the normal phase overwhelming exceeds the superfluid phase by the number of magnums. Because all superfluidities in the range about 0.1 kelvin while we are at 300 kelvin. So the ratio of numbers is about 100. But what we discovered when we analyzed critical velocity and velocity in applied gradient of magnetic field that the velocities of magnum normal part which are diffusion in the force due to very small mean free pass is about five to seven orders of magnitude less than the velocity of superfluid part. And therefore the current will be mainly in superfluid not in normal part. That gives us an opportunity to consider superfluid part separately. But then we have a main obstacle. Main obstacle is that number of particles in superfluid part is not concerned. So instead of normal, instead of usual continuity equation we have continuity equation with right part proportional to sinus of angle. Phi. This N bar is the difference of N q and N minus q. And it is considered because it is Goldstone mode. Now the coefficient eta is very important here because it determines the energy barrier for which you can overcome. And then the motion becomes superfluid but not usual superfluid motion. And this N is C N square over eight divided by H that you see here for a little bit different for symmetric and non-symmetric case. Now what is new in comparison to standard superfluidity? First of all to get superfluidity you should overcome this barrier. So you should submit the energy to your condensate which overcomes some barrier energy which I have written before. Threshold energy. Now imagine that energy, submitted energy is only slightly bigger than threshold energy. Then what we should expect? If energy is small we know that the phase is pinned to zero or pi. Now if we have only small energy or only slightly exceeding the barrier we should expect that it will be pinned almost everywhere and changed by two pi on some solitons of phase change like that. So the phase is on long intervals is almost constant then changes by two pi then again the constant two pi times integer then again changes by two pi and so on. So the change of phases and therefore the velocity concentrated not only in condensate. So velocity is not constant everywhere. So how it can happen? It can happen because the dipolar interaction transfers the magnetization to ledges and then back. Why back? That is because the real processes are forbidden. Therefore you can get on with that. This formula shows how velocity depends on the length or the period of this quasi period of this picture and you see here that the energy is associated with this period and energy of individual soliton. They have different dimensionalities and that is I repeat this formula for. And finally I have shown the formula for the main wall width and the main wall energy and these are conclusions. So please read them. Thank you very much for your attention. Thank you very much.