 Okay, thank you very much in the first place I would like to thank the organizers for putting together such a nice workshop and for the opportunity to give a presentation Today I will talk about the dynamical structure factor of the triangular Heisenberg model a quite old problem And this is work that you know we did in collaboration with Shang-Shun Shang He's a postdoc at the University of Tennessee Yoshitomo Kamiya who is at Rican right now, but he's moving soon to Xiaotong University in Shanghai as an assistant professor My friends and collaborators from the Institute of the Physics at the Rosario, Esteban Joldy, Adolfo Trumper, Mattia Gonzalez and Luis Manuel There is also a long list of Experimentalists that you know collaborated at different stages of these projects and actually provided the inspiration you know for what I'm going to describe in the second part of the talk and And the story really starts with this cobalt based Compound cobalt in this material is in a d7 configuration is cobalt 3 plus So we have seven electrons in the digital that means five in the t2g orbitals two in the issues So we have a spin three halves and an effective l equal to one So out of the spin orbit coupling we end up with a low-energy doublet j equal one-half separated by 250 k From the j equal three-halves doublet and then you know at an even higher energy you have the J equal five-halves so but the important thing is that this gap is two orders of magnitude bigger than the Scale of the interaction between these tablets meaning that you know one can really get rid You know of these higher energy multiplets and end up with an effective spin a half Heisenberg model and because this octahedra. I mean the octahedra that is around each cobalt are you know Have octahedral symmetry to a very good approximation you end up with a very You know to a very good approximation with an isotropic Heisenberg interaction actually DM interactions are Not allowed by symmetry in this material There is a weak easy plane an isotropy as we'll see simply because there is still some distortion and We as I will explain in a moment this is indeed a good realization of a quasi two-dimensional, you know Heisenberg model So this is the effective Hamiltonian for the lowest energy doublet you have as I said an Effective easy plane an isotropy is roughly 10% And then there is an interlayer in exchange that is of the order of 5% times the interlayer interaction So it's a nice Realization of the Heisenberg model in a triangular lattice and actually when you go and do different kind of experiments measuring essentially static properties like the magnetization versus field and Other properties that they don't have to time to summarize in this talk You get essentially the the phase diagram that was originally obtained by and Ray and And Golozov in in this seminar work in 1991 where they use basically a semi-classical approach One-hour expansion and they show that as you increase the magnetic field So at zero magnetic field you have this one known 120-degree magnetic ordering and then as you increase the magnetic field There is some Accidental degeneracy at the classical level that is removed by order by disorder by quantum fluctuations. So these 120-degree ordering basically evolves into this kind of Y-shaped type of configuration for the three sub lattices Eventually you reach Critical magnetic field where quantum fluctuations Stabilize this plateau phase where two sub lattices are up and one sub lattice is down So this this this plateau phase that was predicted in this paper is of intrinsic quantum origin and then You continue with this kind of V-shaped phase The beers in this phase diagram and this kind of w-phase if you want appears in the in this material Simply because of the interlayer interaction. So that is a consequence of the actually 3d structure of this of this model But you know as you can see The the quantum phase that is is really well described by this semi-classical approach and actually one can also Describe the NMR line shape, you know for the different phases that are appearing as a function of field So no mystery up to this point. So the surprise appeared when my colleagues, you know from Oak Ridge Went and measured the inelastic neutrons scattering spectrum of this Sorry At zero magnetic field as a function of temperature. Sorry Yes, you mean as a function of temperature as a function as a function of temperature Yes, so what is the question if there is a phase transition one of Yeah, I mean like for the for the three-dimensional You're talking about the two-dimension or the three-dimension Yeah, really, right? So so anyway So let me continue. So this is the inelastic scattering spectrum and the anomaly that you know, my colleagues notice in This spectrum was an anomalous broadening of the magnum peak of the magnet line around this point of the virion zone This is the end point. So this is a cat as a function of energy at the end point and The only thing they know this is that the width of these peaks was Significantly larger than the experimental resolution. So this was actually the motivation for trying to understand You know first the origin of this broadening if it was real And and the first thing you do is you say well, you know probably this in a semi-classical description This broadening is produced by single magnet to two magnetic case So you go include the next order correction in one over S you go beyond linear spin waves and You try to see if you can explain this broadening in that way and also the spin the bandwidth renormalization that that that is observed because you know, if you Essentially compute with the linear spin wave theory you get these white lines that you know have a bandwidth that is much bigger than than the experimental bang now As I will explain in a moment actually that doesn't work For for reasons that will become clear immediately So first of all with this kind of model You don't get spontaneous single magnet to two magnet decay because the kinematic conditions are violated And that's essentially because of the easy plane anisotropy that you have in the material that although it's small is relevant But you know what actually was more so basically what happens is you do the non-linear spin wave calculation You get I mean you know the exchange parameters here by different means, you know from thermodynamic measurements and also from the long wavelength limit of this Spinway calculation in the long wavelength limit even the linear spin wave is a good approach and you end up with With a bandwidth with a non-linear spin wave theory that is 2.4 million electron volts while the experimental bandwidth is 1.6 So here you have another anomaly, but perhaps the most interesting anomaly is what happens in the high energy part of the spectrum And this was revealed by a beautiful experiment that came afterwards in 2017 the group of professor Tanaka in Japan Where he showed that besides, you know the magnet dispersion that you have at low energies You get all these spectral weight at higher energies that is also dispersive Right so here you can see again a cat at the end point. So this is this point in the brilliant zone So here these are the two magnum peaks at low energy and here you have this broad continuum that according to This group extends up to six times the single magnum bandwidth And moreover when you go and integrate the weight under this continuum It's roughly is more than twice the weight that you have under the two magnum peaks If you go and do the non-linear spin wave calculation at the same point What you get is two very sharp magnum peaks and a very small Intensity in the continuum. So actually intensity in the continuum Turns out to be significantly smaller than the intensity that you have 1.5 times smaller that you have under the magnum peaks So the question is why first of all why this semi-classical approach? You know, it's not explained. It's not producing a large renormalization of the bandwidth that you will expect for the isotropic model and Second why is that? You know, you don't get this kind of broadening of of the magnum peaks that you will also expect for the isotropic model So first of all why in the isotropic limit you do expect some strong renormalization So the reason is that in the isotropic limit you have two types of goals the most you have one goes to mode around the K equal to zero the gamma point and and Two goals to most around plus K and minus K and these goals to most have different velocities meaning that if you start with a magnum That has higher velocity around the gamma point that can decay into two magnums at K and minus K, right? and because of that This single magnum mode enters in the two-magnum continuum You have single-magnum to two-magnum decay because of cubic processes that are allowed in this case because the ordering is non-colinear Besides, you know this broadening of the single-magnum line you get the strong renormalization of the magnum bandwidth that was noticed already in this paper and explained very beautifully in this other paper by Chitomirsky and Sasha Chernichev The problem is that as soon as you put a small anisotropy 10% right you go up out this mode Right you go from having three goals to most having one And now Kinematic conditions do not allow any more to you know for this this magnum cannot decay into two minus here because you know this This will correspond to a very high Energy final state so you cannot conserve energy and momentum Anymore so now you know the two-magnum continuum You get separated from the single-magnum dispersion and you don't have the strong renormalization that you will expect for the isotropic limit It is because of that essentially that you know you go from this picture that this what you will obtain in the isotropic limit You have one you know stable magnum mode and this one that you can see is over damped to this picture Well now you have you know two very sharp Magnum modes and now a small intensity in the continuum here you have a large intensity This again is the isotropic case. This is the real case I mean for that applies to this material where actually you have two things that are playing against this single Magnum this single magnet to magnet decay the anisotropy and actually the interlayer interaction also is important anyway, so Then the question is okay Is this an old and are these anomalies that are observed at zero magnetic field of intrinsic origin or it is disorder or phonons Or something extrinsic that is produced in this behavior So one way of answering that question is you apply magnetic field Right by applying magnetic field you make the system more classical is classical you start suppressing quantum fluctuations and then you expect that if Disorder is not present and if spin of honor interaction is weak Then at some point this Semi-classical approach should start working now in this material Ideally you would like to go to the saturation field but in this material the saturation field is 30 Tesla So you cannot do neutrons scattering in 30 Tesla But what you can do is you can reach with 10 Tesla this up-up down phase the first plateau Right that this collinear is gap So you expect that although it is stabilized by quantum fluctuations as semi-classical approach actually will work reasonably well to Describe the excitations and actually that was I mean it is tricky how to implement a semi-classical approach here Because this phase is stabilized by quantum fluctuations. So you need to Use some sort of trick that was introduced in this paper by Jason Alicia and company What you do is you choose the magnetic field For which the classical phase iron will give you this up up down ordering is one precise value of the magnetic field You do the non-linear spin wave Approximation on top of that point and then you introduce as a perturbation the deviation of the field from that point Right, you will get the gap dispersion. So it will be robust Until you know at some point, you know for some critical value of the field You the gap will close and you will have a transition into this phase or this phase Depending on whether you're increasing or decreasing the field relative to that point One important thing is that Because the interlayer interaction is anti-phoromagnetic the minority spin, right alternates You know from one sub lattice to another sub lattice when you go from one layer to another meaning that you have a doubling of the unit cell along the C-direction So when you go and compare Right the results of this non-linear spin wave Approximation with experiment now you get the beautiful agreement Right the reason why you see, you know, two magnetic lines that are separated by a small energy is because of this doubling of the unit cell along the C-direction and The splitting is small because the interlayer coupling is only 5% of the interlayer good, so Moreover, if you now Change the magnetic field and and look at the low energy branches What you do? I mean what you get is a confirmation of what was predicted in this paper by Jason Alicia and company which is that the lowest energy branches I mean different low energy branches get gapless So if you increase the field, you know, this low energy branch becomes gapless and that signals the transition into this kind of B-shaped face if you lower the field This other branch becomes gapless and that signals the transition into this kind of why Why like face right and you can see that agreement is remarkably good over the whole Field range that covers this this plateau. So bottom line Semi-classical theory works really well. You get essentially the same model the parameters are fixed, right? So we have a situation where Semi-classical theory works really well in the plateau phase 10.5 Tesla But it fails in different aspects at zero magnetic field So that at least Indicates that these deviations that observe at zero field have an intrinsic quantum origin and that sets a challenge, right for you know, how do we describe them? the spectrum of this Triangular lattice system at zero magnetic field and this is the moment where you start remembering that you know actually many years ago Phil Anderson proposed this model as you know potential model for realizing the RBB spin liquid later on Simulations numerical work show that actually the system does order magnetically Although as I will discuss in a moment the reduction of the order of moment is more than 50 percent, right? so So basically what may be happening in the system is that although it orders magnetically it is not too far from Let's say the quantum melting point that will indicate a transition into a spin liquid phase And actually there are DMRG calculations by Steve White and Sasha Cherneshev recently that showed that if you put the second Neighbor interaction that is 5% of the first neighbor interaction That's enough to melt this 120 degree ordering into a spin liquid. That is the MRG results So indeed it is true that we are close to one of these points And then the question is okay if we were far away deep inside the semi-classical regime, you know It's natural to use a gas of non-interacting magnets as your starting point Essentially that is largest expansion and then try to understand the excitations, you know in terms of semi-classical theory, but if if it turns out that you are close to this point Then you know probably it is not a good idea to start with a gas of non-interacting magnets Maybe it is better to start with a gas of non-interacting spinons and then recover the magnets through a Higgs condensation basically of this spinon field and Essentially the magnets will become collecting modes in this gas of spinons that interact via gauge fluctuations So those are nice words. The question is how do you implement that? And you know the natural way of implementing that is through a large N expansion, right? So you can now choose to represent your spins in terms of bilinear forms of this So-called Schwinger boson operators and in order to fix the representation of your spin field You need to impose a constraint that is the total number of bosons here bosons have two flavors up and down because we are working with EC2 and Basically the number of these bosons in each side, you know determines the representation so if we work if we focus on spin what how we have one boson per side and And normally when you implement I mean the original formulation of a row as an hour back If you do simply saddle point or or mean field one of the nice things about this approach is that the the fields that you condense are Invariant under the symmetry group of the Hamiltonian in other words They are singlets both of them are singlets and the rotations and loud and like the usual mean field theory where you condense things that break the symmetry and So basically the order in here in this approach occurs be a spontaneous symmetry breaking be a spontaneous Condensation of these bosons. You are not breaking the symmetry by So and it turns out that you know if you had a bipartite lattice with anti-ferromagnetic Ordering this will be the right operator to condense if you had ferromagnetic ordering on a bipartite system This will be the operator that you condense but you know if you have a hundred and twenty degree ordering You have both ferro and anti-ferro components So it's better to keep both and there are some deeper reasons for keeping both that actually are describing this paper of Rebecca And peers because you know if you want to do a large in expansion in a non-bipartite system You need to work with the SPN group and this becomes the natural the composition So if you want to go beyond the saddle point Approximation because I mentioned that they want to include the interaction between spinons It is better to go directly to Path integral formulation of the problem. So you write your Lagrangian now There is the topological term in terms of the finger bosons plus the Hamiltonian you add the sources that you need to compute Correlation functions and something that is critical in this calculation is you add a small symmetry breaking field Because these bosons when they condense they can condense in a singlet state They can condense in the state that you know actually breaks the symmetry and gives you the hundred and twenty degree ordering and to get that condensate the one that you know corresponds to 120 degree ordering you need to do what is Describing the books essentially put a small symmetry breaking field You first take the thermodynamic limit and then you send that magnetic field to zero Right in that way you make sure that you are condensing the bosons in the states that you know gives you the hundred and twenty degree Ordering and then you will get you know different You know transverse and longitudinal modes and so on and so forth so The rest of the story is basically straightforward You you now introduce a Havars-Trotonovich The composition using these fields A and B that I introduced before you integrate out the bosons And you end up with an action that is a function of your Havars-Trotonovich fields The phases of these Havars-Trotonovich fields and the chemical potential sorry here You know this basically chemical potential is introduced here to enforce the constraint of one particle per side Those are basically the gauge fields of your theory that are coupled to this pin-on field and In order to develop some diagrammatic expansion of this problem Your basic elements are the spinon propagator at the saddle point level The RPA propagator that is essentially the inverse of the fluctuation matrix that you get If you do a Gaussian expansion around the saddle point in these Havars-Trotonovich fields And then you have external vertices associated with these sources J that you need for computing, you know correlation functions and you have internal vertices that you know give you the Interaction between this matter field the spinon field with with the gauge field So At the saddle point level, we know that we are going to get a result that is qualitatively wrong Right because at the saddle point level our spinons are non-interacting. They are free So basically if we compute them at the dynamical susceptibility What you will get is you know, we insert a spin excitation that spin excitation will decay into two spinons, right? The spinons will reconnect and come out Meaning that what you will get is simply a two spinon continuum, right? Branch cut singularity, no poles, no magnets And this is the reason why no one really likes to use this approach At least at the saddle point level to describe, you know, the magnetic excitation of a magnetically ordered system You need to go beyond saddle point So now if you go beyond the saddle point level You get these four diagrams to order one over n the way you compute the order of a diagram is You each wave line each RPA propagator introduces a one over n and each internal loop like this one Introduce a factor of n so you can count and see that you know here You have only one wave line one over n one wave line one over n one wave line one over n Here one wave line one internal loop Basic it's two wave line sorry and one internal loop again one over n now in the rest of the talk I will focus on this diagram and the reason is that this is the only diagram that can introduce poles in in the In in in the s of k and we are looking for magnets for poles And you can immediately see here that this RPA propagator is evaluated the same k and omega as the external line So poles of the RPA propagator will immediately become the new poles that you get in s of q and omega now that diagram Right, you know, it's interesting to to see what happens if you Compute that diagram on top of a condensate that doesn't break the symmetry, right? So if you condense your spin-ons in a singlet state This diagram contains this loop this loop is a cross susceptibility between an external line and an internal field As I explained before the external lines, of course are basically spin components, right? They transform like vectors The internal lines are these have a straight on each fields that I said the air sky scalars So if your ground state at the saddle point level is a singlet state The expectation value of this bubble will be zero right so this diagram will will cancel Now this is the reason why it is important to put this symmetry breaking field because you know Once you have a spin-on condensate in this broken symmetry state this diagram this bubble actually becomes non-zero This diagram then Becomes finite and when you go and compute so you add this correction to the saddle point Level so you get this comparison. So this is what you get at the saddle point level This is the component of the dynamical spin structure factor in the direction perpendicular to the plane of the hundred and twenty degree ordering While this is The component in the plane, right? You have the two directions are equivalent in that plane So you get here I'm plotting the average of those two components And this is the sum of of these two the sum of the three is the trace if you want of s of q and omega Now there are several disturbing things at the saddle point level So I mentioned already one the fact that you know these lines actually are not poles are branch cut singularities That you know correspond to the lower edge of this to spin-on continuum But moreover you get these porous modes, right and physical modes That you can tell that they are unphysical because if instead of computing the s of q and omega you compute the density density correlations You get a maximum at these modes and you know that density should not fluctuate in this approach, right? You know the constraint is that you should have one boson per site So you know that these modes aren't physical So these are the two reasons why you know people try to stay away, right? You know from this kind of saddle points if you are describing a magnetically ordered seat The nice surprise is that as soon as you add these Gaussian fluctuations this other than first of all this spurious modes cancel exactly The second thing is that this maximum that you get at the lower edge of the two spin-on continuum right this part disappears and Moreover, you start getting your magnum dispersion, right? You're collecting modes at low energies right here. You can see it more clearly the magnum dispersion Right, so now you recover your magnum modes and you eliminate all these porous effects That you get at the saddle points and to show you in that in more detail You know here. I'm showing the saddle point result This is you know spurious mode plus the bottom of the two spin-on continuum You can see that the one over end correction cancels exactly these contributions and now The new thing is a new pick that you know is a pole now It's a magnum in s of q and omega, right? So you get the qualitative improvement when you include this correction now Although this story is all motivated by the high energy modes by the fact that you know Tanaka observed this continuum of high energy modes up to roughly six times the magnum bandwidth Sanity check of the story is to go to the long wavelength limit now and try to see if you know this reproduces what You get from spring waves that you know that it should work in the long wavelength. Yes Right, so it's it's one over end correction is this blue line, right? It's the diagram that they showed before right the wavy line You don't need to put any to put any particular value of n at this point You are summing you know contributions coming from you know diagrams that appear to different order in one over I don't need to specify n at this point, right? Well if n is equal to infinity, then you know you will get you know Basically you will end up with this but if n is equal to infinity, then you know this other point also doesn't make any sense Basically here it appears The cancellation seems to happen, you know for any value of n, right? So it's not something that you know you need to put in in advance and how I Mean we can discuss it later, but you know the other Interesting thing is that you know if you now compute the In the long wavelength limit you compute the single single magnum dispersion You get of course the linear dispersion, you know you get the your custom modes and you get some velocity is You know around the gamma point is given by this value with Small decay that this is proportional to q squared So now you can go and compare right this a spring wave velocity versus You know the spring wave calculation actually with a non-linear correction and You know what you get is that you know the both around the gamma point and the plus minus k point This spring wave velocities actually turn out to be very similar. So somehow you reproduce pretty well the You know the hydrodynamic limit that as I said, you know, you know, it's well described by the one over as expansion so the other Check that you know one can run is simply compute the order of moment You can compute the order of moment by taking the derivative of the ground state energy as a function of this symmetry breaking field and You know you can do this calculation different ways You know you have to be careful because you know here you have zero modes associated with this gauge invariance So you can either use for that pop off or you can truncate your fluctuation matrix in both cases You get the same result What you get is that the order of moment is 0.22 the full moment is 0.5. So basically This number is quite close actually to values that we're obtained, you know with different numerical approaches This is density matrix normalization group. This is quantum Monte Carlo and this is serious expansions If at the cell point level you get 0.28 and with no linear spring waves you get 0.25 The other check is, you know, you can go and compare the magnet dispersion that you obtain with this Basically saddle point plus fluctuations against the result of serious expansions. This was done in 2006 by Rashi Singh and company and What you find out now is that the bandwidth is Roughly 20% bigger the bandwidth of the series expansions is 20% bigger than what you get with this, you know one of an approach The okay, so keeping this in mind now, you know, it's time to go back to the actual compound So far I have described, you know, what happens for the for the 2d limit for the 2d Heisenberg model And you know in that case you have to introduce two additional basic Havart Trotanovic Fields When you do the calculation back to this compound you basically get Now a dispersion a single magnet dispersion that has a bandwidth of 1.3 millilectron volts While the experimental bandwidth is actually 1.6 So you underestimate roughly also by 20% the experimental Finally, I mean in addition to the two magnet modes I mean here one of the magnets is over damped you get this high-energy continuum here That corresponds to the spin to spin on continuum and you know may be connected to this high-energy Continuum that is observed in Tanaka's experiment Actually, the the the experimental continuum extends up to you know, higher energies even higher energy Higher energies than the two spinon and finally So to conclude, you know, let me say that you know one can reproduce pretty well at least in this qualitative aspect of the High-energy part of the continuum of you know of this material Using this one over n approach But you know keeping going beyond the saddle point level the magnet dispersion is roughly 20% lower In bandwidth that what you will get You know with you know in the experiment You get a qualitative improvement related to saddle point right because you get this removal of of the spurious modes and then In addition you get this extended to spinon continuum You get correct order moments and magnum velocities in the in the long wavelength limit and something that you know I didn't have time to discuss which is that you know, you can reproduce the You know, basically if you take the large s-limit in this approximation So in the constraint you put s very large you get exactly the result of spin waves So basically you reproduce spin waves in the large s-limit Thank you for your attention