 So come to the second talk of the session by Aleksaaric University of Pugna on the academic theory of the entire Kinspin in the picture. Okay. So, sorry, five minutes. Okay, good. So, you hear me? All right. So, I would like to thank organizers for inviting me to speak in this nice place where I learned so much, so many interesting talks. So I will be talking about a very simple but I believe still interesting story of which connects to the big question of spin liquids and the response magnetic field. But before I do that, I would like to take a moment to advertise KTP program for the next summer. A new spin on quantum magnets, which is organized by Christian Batista, Natasha Perkins, Lucille Feller and myself, and the deadline for applications is October 2nd, and time moves fast. Okay. So this talk is based on a series of papers which I have done in collaboration with Leon Balens at KTP and Anna Kesselmann at Technion and my former student, Ren Bovank, who just moved to postdoctoral position in the University of Montreal, and a group of ESR experimentalists, Kirill Povarov and Andrei Zhevodyev from ETH Zurich and Timofey Saldatov and Alexander Smirnov from Kapitsa Institute. And so at the end of this talk, I will show comparison of our series of experimental data obtained by these people. Okay. So the outline is such that I will give brief and very general introduction to kind of the benefit of students of what spin liquid is and fractionalized excitations on a very simple level and then move on to make a claim that putting a spin liquid in a magnetic field actually allows in a way, enhances interactions or maybe reveals effects of interactions in a very simple but profound way. And then I'll focus on one dimensional very simple well understood spin liquid where I believe we discovered something that has been overlooked for many years and compare with experiments. Okay. So, so the big motivation for this talk and this line of research is really search for spin liquids, which as you all know, I'm sure has been going on for incandescent matter for last 30 and actually probably longer, more years. And so the spin liquid is a state of magnetic matter which breaks no symmetries and is characterized, it's a very quantum states as a ground state of a many body quantum system. And it's an entangled state. And as a result, it's excitations are known to be fractionalized that the carrier fraction of the usual quantum number of magnetic excitations. And I'll say a bit more about that later. And interactions between them are mediated by gauge fields which are emerging in the theoretical description of these things. And this is a very exciting and highly motivating proposal. And over years, there were a number of experimental candidates, most of which have failed the stringent test of compare the new theory and turned out to be a magnetically ordered materials. But we are still hopeful that this is not a theorem that forbids quantum spin liquids in nature and we will find one. And then the question, actually, big question is how do we know that we see a spin liquid when we find one? So what kind of experiments we do that convinces us and also our referees that it is a spin liquid. And this is a non-trivial question because, well, experiments are complicated. And there is no universal answer. And I will certainly not, I don't know of one. But again, focus in this talk will be on adding magnetic fields or studying response and magnetic field. And this is not the unique, but one very useful way of addressing this question. But so that's a very kind of big picture. So let me, for the benefit of students, very quickly remind you kind of key differences between so magnetic solids and magnetic liquids. And so we are talking about very simple Hamiltonians, well, simple to write, difficult to solve, which are essentially of exchange kind. And I take them to be symmetric. This is restrictive, but is, and we know that everything, most materials in nature are not isotropic magnets, there are always corrections and they are important. But for the sake of argument, let's start with the simplest Heismarck model. And then in the audit state, the wave function is essentially a product state. So for example, for this triangular antifuramagnet, which is sketched here, which was already discussed several times in this conference, the ground state is 120 degree structure of spins. And you can, you know, the wave function you can write as a product. So introduce three sublattices to in this triangle lattice and on specified direction of spin on each of the sublattice, and that's your wave function. And then excitations, spin flips, block waves, which you made of a spin flip, propagating an appropriate background. So I showed 120 degree structure. This is not it's much simpler structure, which is easy to draw. But the magnetic excitation is a flipped spin and it carries spin one. And it, so in neutron scattering experiment, one sends a neutron, which interacts with flips a spin and emits or absorbs a spin wave as a result. And for a given momentum, there is a well defined frequency. So it's a well defined particle. We can study and people have studied questions of our line beats and such, but at the zeroes approximation level, it's a delta function like quasi-particle, which is seen experimentally in neutron scattering, for example, as for a given fixed momentum, there is a response at a fixed frequency. And so that's, that's simple. And that's how it has been for some time, even though the history of the subject is highly non-trivial and not straightforward, but in late 70s, and then again at the end of 1980s, Anderson made a profound suggestion that another type of ground state is possible fully quantum entangled, so with something which we call now spin liquid. It was called the Rhythmic Invalence Bond, where instead of defining magnetic moment on sub lattices, you imagine, for example, putting pairs of spins in singlets and covering singlets lattice with the singlets. And the idea, you know, comes from essentially chemistry, benzene molecule has been analyzed by Pauling, sorry, you know, before, and that's a profound state of the finite quantum system, which is a benzene molecule is, and, but so Anderson have taken it to thermodynamic limit, and of course, gave much more thought to that. And so what's, if we imagine, for example, so such a wave function, so such a quantum right state, which is characterized by all possible pairings, the singlet coverings, okay, so this is a highly non-trivial and entangled state, and a very key feature, physical feature of this entanglement is that when we try to create an excitation, and we create excitation by breaking say one bond, taking it from singlet to a triplet state, as shown here, okay, and then by acting with Hamiltonian on such a state, we can separate this individual spin halves forming, making a former singlet that has become triplet into a pair, so this pair can be separated infinitely far away, okay. And so as a result, we have an excitation called a spinon, which carries spin half, okay, and that's a fractionalized excitation, okay, and the line of flipped or displaced singlets shown in green essentially is a cartoon picture of the gauge field, which we get interaction between these, and so the key physical property of the spin liquid state is that its elementary excitation carries spin one half, and they're always created in pairs, so any local operator acting, so spin flip or group of spin flips in a finite small moment volume, sorry, sub volume of the system creates even number of them, and this is, has profound consequences both for theory and experiment, so this slide I just, so these are all very cartoonish, very simple pictures, I just want to say that in last, let's say 20 years, huge progress has been made in understanding the spin liquids, and we have several kinds, theoretically, we have several kinds of liquids, and for which many of which have been confirmed by, you know, high large-scale numerical calculations on model Hamiltonians, so we do know that we can cook up a Hamiltonian, which will have such a ground state, and the question is now if, you know, if we can find materials in nature exhibiting these properties and study it, but out of all, so there are many kinds of spin liquids, and my focus in this talk will be, you know, is highly motivated by one probably the most exotic, or at least the most counterintuitive of this, is a so called spin liquid with spin on fermion surface, so this really is a simplest way to think about is a Fermi liquid of neutral fermions, so the spinons are fermions, they don't have charge, but they have spin one half, and the spin response of such a spin liquid is determined by the response of spinons interacting with gauge fields, and so now I will start to kind of moving towards a more specific description of what I mean, so I'll describe the background, and so again this is a spin liquid state, which I would like properties of, which I would like to understand, and if possible compare with experiments, and so the fact that elementary excitations carry a fractional spin one half means that very basic fact, and very important one, is that the neutrons get an experiment similar to one I previously shown, when neutron creates a spin one excitation, that excitation breaks necessarily into a pair of spinons, maybe not one pair, it can be, it can break in two pairs or three, but the leading process, unless something Hamiltonian is very special, is breaking into two, in a two spinons, and as a result momentum and energy of the neutron, the transfer to the material, is shared between these two spinons, and as a result, for a given momentum transfer, if you look on this, so the pictures are kind of small on this screen, so if you look, so what is shown here is experimental data, neutron scattering data from one of the one-dimensional materials, and so horizontal axis is momentum transfer, so the k, and the vertical axis is omega, the frequency, and so since Yosh, the total k is made of the sum of two, moment of two spinons, and the energies, so that energy and momentum is, conservations occur, it means for fixed total k there is a range of energies at which you observe response, so what you see is a continuum of spinon excitations, and that continuum is characterized by shape and by intensities, and for example, here for some given momentum transfer, you see a line shape, experimental line shape from that material, which is highly asymmetric, non-trivial, extends to energies, very large energies, so this energy range in terms of exchange integral of the underlying magnetic Hamiltonian is several times j, roughly, so it's huge, so it's not a witz of the order of some fraction of j, but it's several times, and then here is another example of experimental data from two-dimensionals, spin-half antifreeze magnet, more recent data, and it's kind of great to have this data, but it's always a puzzle, you look at this and try to make a case whether what you see is something which approaches spin liquid or it's some ordered magnet, but which is disordered either by disorder or by some interesting interactions between magnets, and so that's essentially a question which always is present in this field of research at which point, how, what kind of data do we need in order to, again, make a convincing case for the spin liquid? Okay, and so let's, so I again argue that magnetic field helps to understand and sharpen at least some of those questions, so let's quickly look on what magnetic field does, and I will use an analogy with Neutral Fermi liquid or even Fermi gas to first kind of explain basic features, so we see, so we apply magnetic field and we split our spin and bends for spin up and down, spinons, okay, so by Zeeman energy of course, and so, and now we ask, so we experimentally will, you know, interested in studying transverse spin dynamics in a system which is placed in external magnetic field, so we transfer as pluses minus, so we create excitations with spin one, total spin one, and we can do it a number in this simple picture in of course many infinite number of ways, but the simplest one, for example, is a vertical transition shown by this red line going, you know, flipping spin up to down and vice versa, and this momentum does not change, energy change exactly by Zeeman energy as it should, so this point on energy momentum plane, or we can take a transition, zero energy transition, but between different Fermi points of up and down spinons, and this doesn't cost us any energy but requires finite momentum, which is this point, another point on the hue axis, and so these two points kind of define this triangular shape wedge within which you have a particular whole response of a spin on gas in this case, okay, and importantly magnetic field frees up this little corner, this little triangle, the white one, where in the gas there are no particular whole excitations, so now of course we want to understand system use interactions, so we add the simplest possible interaction, something like a local interaction which respects symmetries, and in magnetic field there is a very simple effect when simply essentially mean field really effect where you ask, so this interaction creates a molecular or internal field, so the spinons with spin up feel magnetization, essentially a polarization created by spinons, we spin down and vice versa, and just on second this, and so in magnetic field these polarizations for up and down are different, and as a result out of this you get a term like that where u is interaction, m is magnetization, and so as a result the total field that spinons with up and down spins experience now some of internal plus this external, sorry, and this internal field, and as a result your continuum shifts to from Zeeman energy in this simple naive picture to the energy at q equals zero to Zeeman plus u times m, there was a question, it shouldn't be linear, so it's linear for small q, but no it's for the ease of drawing, yes, so for the gas it is, for with interactions it's not, yeah, yeah, so absolutely, so with interactions this thing is filled with, but the spectral weight is much smaller, weaker than inside the continuum, yes, yes, but okay, but so the point of this very simple picture is that this kind of correctly looking procedure which looks correctly correct maybe at zero, so even minus first order is actually profoundly incorrect because we have to recall something which is we know from undergraduate studies Larmor theorem which is saying that at q equals zero, zero response of magnetic system provided Hamiltonian is symmetric, is right, commutes with a total spin is determined only by external field which is small h here, okay, so you just commute total spin and if Hamiltonian is a symmetric aside from the Zeeman magnetic field piece, that's the only piece that contributes and as a result, so what Larmor theorem is saying is that q equals zero all response must be at the value of non-renormalized Zeeman energy and so which, but I just explained to you that continuum shifted up by u times m and what it means that there are additional interactions and in this particular case it's very easy to calculate the ladder series, they recover this property and one finds by summing this infinite ladder series which describe interaction between spin up and down densities as shown by these wave lines, one obtains interactance susceptibility which contains a pole in addition to the particle whole continuum of spinons, there is now a collective mode shown by red which goes under the name of ceiling spin wave because it was actually originally discovered by ceiling at the end of 50s and this mode in metals for metals and this mode was observed experimentally in 1967 and I just want to make a comment that at the time of the observations this was taken as a experimental proof of the Landau Fermi liquid theory validity of Landau Fermi liquid theory for the metals because this collective mode only appears due to interaction, it's effect of interaction, you shift up continuum but interactions bind in that case in ceiling's case fermions to make a collective mode with spin one that mode disperses downwards and it has very nice for experimental purposes property that at q equals zero all spectral intensity sits in that red branch and as you move away I find at q spectral intense some of it shifts to particle whole continuum so it's a sharp well-defined peak so to Nikola's question so there is finite lifetime to this mode due to higher order processes but it's small and moreover again the same Larmor theorem saying that in the limit of q going to zero lifetime becomes infinite because you have to respect as I see yes no it's it's valid everywhere yes sure yes what is what so the so if there are is there is a spin response so it's a precession of magnetic moment so if your insulator has magnetic moment it processes you you see it one way or another I said the Hamiltonian is symmetric yes yes yes so spin orbit interaction will be important for later part of my story but yes yes okay so so that's very simple and so now okay so first of all so that's that's a still a warm up so arguing why so that in magnetic field so if we now imagine spin liquid spin on thermo spin liquid of the spin on thermosurface kind this physics directly applies there okay so there should be when sub when such a spin liquid is subject subjected to external magnetic field the smoke you and energy continuum restructure it dramatically and instead of incoherent particle whole like response one should see a sharp spin one collective mode with a you know dispersion which we can figure out we so thinking of such a spin liquid one can add effect of gauge fluctuations so which also contribute to lifetime we did that but I will not focus on it here I'm happy to discuss it but you know it doesn't change the basic physics what I'm describing okay so so okay as I just said would be great to see this physics play out in experiment and here we okay have to face an issue that while we continue hunting for spin liquids you know the most reliable most most reliable spin liquids that we have and you know understood and have you know in some cases a very good experimental control of our one-dimensional so which is not to say so there are some very promising materials and I just mentioned them because I you know like Kagome antifuramagnet hypersimicide is still a mystery as far as I know so there are of course now Ketayev materials which are extremely interesting but to which are the opposite limit of extremely spin orbit coupled materials where this story doesn't apply directly and there are some actually spin half triangle antifuramagnets which are look extremely promising they're two-dimensional and quasi two-dimensional look extremely promising and there are some neutron data appealing on them but let me take so focus on 1d which because that's where the experiment which I aim to present to you has been done and so again I'm showing here a couple of examples of neutron scattering on yet different one-dimensional material so the top panel is experimental data bottom panel is theoretical I believe there is single fitting parameter with exchange energy and then you know the colors are intensity so okay there is good understanding non-trivial line shapes for given momentum again line shapes are extremely long interesting and delicate and it's a very interesting and deep theoretical problem to understand exactly singularities you know exponents which govern these line shapes and their properties but as I said and a lot of research has been done by many people in the room including you know many glasman alexa et svelik and others but those singularities at the same time you know experimentally most difficult to observe so nonetheless I will argue that placing simple Heisenberg chain in magnetic field leads to features which allow us to directly access interaction between spinons and analyze it so let me kind of summarize key properties of one-dimensional spin half Heisenberg model so it's written here I so we have exchange interaction between neighboring spin j1 and therefore for future use i at j2 so connection next nearest spins and in general we'll have magnetic field and so a lot essentially is known to this about this model and in particular so if we look on the ground state sorry phase diagram zero temperature as a function of j2 to j1 ratio so so when this you know finite range from zero to essentially a quarter j2 you know to j1 of 0.241 we have gapless lichendro liquid critical ground state if we increase j2 beyond this point it becomes dimerized spin gap opens so in the field so proper or convenient so there are several theoretical descriptions of this simple Hamiltonian and I will use something which is called based on spin algebra and so because it most directly connects to to high dimensional spin liquids and in this description we have two sets of Dirac fermions Cyra R and Cy L which describe excitations near right and left fermi point okay we linearize the spectrum so these derivatives just linear momentum and so size stands for these fermions so which are spinons and then we have additional in theoretical terminology marginally irrelevant interaction which couples right and left spin currents which is j's with index r and l and j's are really just magnetization densities and right formed at the right and left fermi points okay and the original spin at a particular point is sum of right and left spin currents spin current is a technical term so there will different from magnetization current you just terminology and then it's the chain is anti-framagnetic so there is a staggered Niel vector which is a combination of left and right spin is okay and so this is the main interaction term and it's marginally relevant which means that as you take as long as j2 over j1 less than this critical value of 0.24 as you go to low energies and you know long distances the this coupling constant g vanishes progressively vanishes and in particular if you look in the ground state of the infinite chain limit it's simply zero but it flows to zero logarithmically weakly and that's the marginal in the marginally irrelevant and that happens to play a very important role in what I say in a few slides and show to you and so this so with rg scale l for those who like this way of thinking you know g of l vanishes as essentially one of a small l which is lag of energy but in the presence of magnetic field this flow which is known as castellate stowell's flow is cut off and so that at finite magnetic field instead of going along the diagonal and in this little diagram or one you know the the rg flow changes so that interaction between z components of spin currents which are vectors you know is finite but interaction between transverse components vanishes but we'll be we are interested in looking at the response of this one-dimensional spin liquid at energies of order Zeeman energy and so our coupling constant g is finite in that region and that's important okay so okay so that's and one last sorry thing to add this initial value of the coupling constant g is function of the ratio j2 to j1 and so for pure Heisenberg chain it's something finite and positive in the way I defined it and as we increase j2 to this point 24 value it at this point it vanishes at that point system has no so the leading marginally irrelevant operator is absent and the system is described by pure conformal field theory with no perturbations for yes it's zero temperature yes zero magnetic field yes and so now we add magnetic field and kind of so this is again thinking of non-interacting spinons first so you have this transition so on these two plots so if you look on the right most it's more contrast easier to see so you see the transverse spin susceptibility of non-interacting spin chains so chain in which I took back back scattering constant g to zero by hand then of course it's a very simple calculation and so you again see this triangular wedge within reach there is a spectral weight even though in one dimension with Dirac fermions essentially all of the spectral weight is a delta function like concentrated on these boundaries but okay for generality we know that there is more okay so this non-interacting picture has been you know extensively used because it's extremely good approximation and if we zoom in and this is a picture the same picture taken from the kind of classic paper for Chicago and Affleck shows this again triangular wedge so at zero momentum there is single crossing point and the first surprise comes when kind of you think or rather even dig through the literature looking for numerical data and then we find that numerically transverse dynamic structure factor of the spin half chain in magnetic field has been studied multiple times yes so I'm I'm afraid to do this experiment in real time oh view options okay I did that so is it better or do I need to I did stop sharing okay so here yes okay and then view top rise sorry I didn't see you very nice okay all right so you can't see it because it's on the other screen it's on this no no you're on probably this one if you if you can thank you so okay so all right so so the point is there is a finite um splitting uh abstract numerically uh so uh just look on top row this is transverse structure factor for different magnetizations and as magnetic field increases the splitting increases okay and so the first question is you know what what is it and and uh of course the intuition which I was trying to present previously thinking of non-interaction uh neutral fermions can be used here so so we the only interaction is back scattering which I now have written here explicitly z and transverse components plus and minus and so uh proceeding in the mean field like sinking uh which turns out to be surprisingly uh you know good description in for this particular problem uh we can replace uh longitudinal components of the spin currents with expectation value which is just half of magnetization because some of them is magnetization and then our continuum as shown by dashed lines here again shifts along frequency axis to the renormalized value just as in high-dimensional uh non-interacting gas that I showed you and of course again the larmor theorem is saying this cannot be uh uh correct and we have to account for the transverse interactions and we've done it uh this can this can be done in a number of different ways uh but uh the most straightforward one which is uh heterodynamic like uh which is uh uh where uh we use utilize this uh cut smoothie algebra that uh the spin currents obey uh so it commutes you know the commutator is not only so the generators of rotation but the commutator is not only the third component but also this anomalous piece derivative of delta function so that's kind of essentially field theory input uh in this case and then derive dynamics equation of motion really for magnetization m which is sum of spin currents and uh magnetization current l which is the difference of current okay and the these are two exact equations so as long as Hamiltonian with linearized distortion you know is a good approximation so that's the exact equation and uh interaction only comes in this term m cross l so this little g is a back scatter interaction okay proceeding in this uh way we say that obviously in magnetic field there is finite expectation value to m we replace it so there is a delta which is dimensional is the most important dimensionless parameter of the theory uh so interaction parameter so g divided by pop i v okay uh so delta enters in denominator because we account for molecular fields produced by internal fields produced by the spinons after uh that simple approximation these two equations become linear and what comes out explains 95 percent of the puzzling numerics uh which is that now we have two branches so instead of dashed lines look on these solid lines so there is a finite splitting which is determined controlled by interaction this dimensionless delta at zero momentum uh the downward branch disperses down just as in high dimensional uh ceiling wave the upper branch disperses up uh and the larmor theorem is satisfied in the following ways that uh residues the so if we calculate susceptibility so mod plus and minus come with residues a plus and a minus and larmor theorem is satisfied that q equals zero uh that uh of the green modes uh a a a plus uh vanishes okay so at q equals zero you see response only from the total spin precession but at any finite q you see two modes and the splitting and we have you know so simple equations describe dispersion and the splitting between these uh bands and uh this has been okay the two minutes okay this has been confirmed uh numerically uh indium rg uh and so uh let me really okay so again numerical data by uh ana uh keselman uh so we are looking magnetic field and along horizontal axis plotting j2 to j1 so for Heisenberg chain you see the splitting for given field is maximal as we increase j2 to the critical so this conformal point where back scattering vanishes that splitting vanishes and so we can feed all these lines by the simple series that I presented to you so let me now probably in two minutes formulate uh what has been done experimentally so there is a so if we have such a ideal Heisenberg chain subjected to magnetic field as I just explained to you yet and we do esr experiment which is exactly that you subject system to magnetic field and you pump it with microwave radiation and observe our energy absorption so and this experiment measures really vertical transitions because wavelength of the microwave is huge uh bigger than sample sizes and so you really prop q equals zero if the chain is ideal you get just exactly you know response at the human energy and you learn nothing about the system no no matter how strongly interacting spin on the we are saved by an isotropy by uh spin orbit interaction which in that in case of particular material which is uh shown listed here uh is described the leading spin orbit interaction comes in the form of uniform Jolashinsky-Marie interaction so this term s cross s with vector d which is a spatial vector determined by the crystal structure of that material and important point this interaction doesn't uh break a translational symmetry and if you the essence of the why it's useful is that already on the lattice level you can for the special geometry when b is oriented along magnet Jolashinsky-Marie axis d you can perform a position dependent rotation unitary transformation of your spins so as to remove Jolashinsky-Marie interaction from your Hamiltonian but the price for this is that since unitary transformation is position dependent you give a momentum boost to your momenta so all momenta wave vectors are shifted by d over j exactly okay and so therefore measurements done on such material at zero momentum are equivalent to the measurements done on the ideal spin chain at finite momentum d over j and so in this way ESR provides access to small q region of the dynamic structure factor of spin half chain okay and experimentally I just show the results so they so eth and capyta group have done exactly this measurement and what they observe is best if you focus on this middle plot so we have two branches so it's a we just subtract paramagnetic gamma h line so there are two branches corresponding to omega plus and omega minus that I essentially I was showing horizontal axis is now magnetic field right the momentum at which we probe is set by Jolashinsky-Marie okay is magnetic field and points are experimental data and solid lines is a fit to the theory and the fit involves single parameter delta interaction strengths which comes out to be point 12 okay the splitting is determined by Jolashinsky-Marie vector determined at zero or very in the limit of finish and magnetic field and there's another parameter which is extracted directly from experiment so in this this chains are characterized by exchange of order 20 kelvin and G Jolashinsky-Marie is 100 times smaller okay uh visible in ESR once this is done so we determine the interaction which I rename to you just because not to be confused with g factor as experimenters ask and so and this is dimensionless back scattering interaction at the energy of evaluated at frequency which is magnetic field uh set by magnetic field okay uh in addition experimentally the intensity of zappa mode vanishes relative to the intensity of the lower mode if we take the ratio we get rid of all unknown factors and here is this plot ratio of intensities which is parameter free and so you see the this going through exactly what we predicted and more over the last point and I stopped here so I was telling you earlier that the coupling constant g or delta in my annotations here now is running flows under g and so and for ideals Heisenberg chain it's known exactly so almost exactly uh meaning uh so this result uh so gives implicit equation of delta as a function of temperature and magnetic field and we know j we know temperature we know magnetic field we plug in and we get uh this different curves which are right in the middle you know point 12 is right where it should be and so this uh experiment and its theoretical analysis represents to our knowledge the first ever spectroscopic determination of the back scattering interaction in Heisenberg chain and we are okay and so at this I uh we have many more uh data but I should stop here and thank you are there a question okay thank you can you explain a little bit more about what is the effect of the Gelochinsky Maria in your picture yes with the magnetic field what what happened how did the Gelochinsky Maria modify that picture so essentially so yes so Gelochinsky Maria okay so introduces essentially becomes in this formulation uh allows us to access finite moment so momenta by this unitary transformation which outlined here so so really okay so we have Heisenberg term which wants to spins to do this and Gelochinsky Maria which if that's the axis it wants them to be in the plane okay so classical solution for example of this so slow spiral with a pitch which is determined by exactly DOJ right so quantum mechanically so how we see it that if we do this simple unitary transformation uh your Hamiltonian transforms into this written in terms of spins with tilde which is almost not quite and it's important but uh Heisenberg term and Gelochinsky Maria disappeared but all the transforms spins come with a factor e to i x times DOJ so which means all momenta shifted by this amount okay oh so so you can think okay so you can think by analogy for example this uh Rajba or some other spin orbit it's internal field built in which right and left movers which is opposite for right and left movers sorry yes it's a carl field which uh which couples with opposite signs to right and left movers and this provides a momentum boost that's another way to think of that momentum boost and which then turns ESR probe into which is q equal zero probe into probe of right and left moving uh spin spinants sorry in this chain oh so so so the ceiling spin wave yes uh so uh so in magnetic field particle whole continuum shifts well as so for small momenta transfer it shifts up in energy and uh so the particle whole continuum response sorry uh exist uh within this shaded area mostly uh as uh was correctly pointed out and then the if that would be the whole story then larmor's theorem would be violated because look just looking on this shaded uh cyan uh you know picture you you'll be forced to conclude that at zero momentum system response at the frequency which is z one plus interactions sorry times magnetization and this cannot be true uh as was recognized by ceiling and many uh others after him and uh instead uh interaction between spinons or electrons and holes uh in the case of metal uh produce a bound state with spin one and this bound state uh determines the response at small momentum what can you say about the the line shapes of the if you are in other words uh it shouldn't be just simply delta functions right they shouldn't be yes and uh there must be exponents associated with those so so right so the so the line width roughly scales with temperature linear temperature and we understand that that comes from that the fact that after unit transformation x y part of the exchange is slightly different from z part so there are squared corrections which lead to this lifetime but uh uh okay so i didn't yeah but there should also be i mean these things should be some kind of power laws i mean i so in this particular case i believe it's temperature so there is no non-trivial power law we are talking about small q response so this really so there's it they don't show in other words you don't see any of the so so this for i mean you know this is an example of line shape actual line shapes so it's so it's it's was pretty difficult for my collaborators to extract the positions of the centers so the the line shape is you know they often overlap so for example if you look here so it's a very good question and specifically i mean you could do instead of doing c w you could do uh pulsed epr and then the question is what is the decay what with the t1 and t2 decay right okay so that has not been done yet so that what is it what it for the theory would predict some kind of our law i wouldn't predict the normal so i think uh so to to lead in other interactions i think it will be uh simple power laws but let me uh that's my expectation so there will be a correction okay i don't see any other questions so let's thank the speaker again we'll have a coffee break and then come back for the last session