 Yes, no. Okay. Okay. Fantastic. All right. So good morning, everyone. I'd like to discuss magnetic exchange and exchange restriction from Vania 90 and DPW. This is the work that has been done by basically a number of people from my group. I'd like to particularly acknowledge Ryo Tawono, who was there at the conference. Also, you had the chance to interact with him and Ditya Putatunda contributed to some of the results. And then some of the results that I will be discussing also resulted from the work of Francesco Fajetti, Makarita Parodi, Ravik Kaushik, and the former colleagues, Uipanet and Pongchand, who are now, who finished basically in this group and are now working elsewhere. I'd like also to acknowledge senior colleagues, useful discussions with David Vanderbilt and Feliciano Giustino. So the motivation for my talk is that in many situations, the systems that are most interesting, where interesting magnetic states and transitions are observed, are the frustrated ones, where the exchange interactions compete. Especially when non-trivial exchange interactions are there, the non-isotropic ones such as Isink and Kitaev exchanges that are induced by strong spin orbit coupling, as well as the Lashinsky. And so on this slide, I show a unit cell of Niki Uteurite with material with different fields, as a range of different magnetic states. And the magnetic Zeeman energy is actually the lowest energy scale in the system. So you see this lowest energy scale by just switching magnetic field a little bit. You can switch between those closely degenerate states and get all these different spin states. So in this system, to reproduce the correct ground state, one actually needs to include five different nearest neighbor exchange interactions. And so what is crucial to understanding the physics of this material, it's a giant magnetic electric effect. And this material has the largest magnetic effect of single phase compounds. So following the suggestion of David Vanderbilt is looking at the barrier between the two different states. Turns out that the force order anisotropy, so single line force order anisotropy cosine force phi, is basically responsible for whether the transition is going to be continuous or first order. And so basically, when the magnetic field is ramped up or down in the center for magnetic material, a spin flow transition happens. So the transition between Cta equal zero, so spins vertical and Cta equals by half, where spins are perpendicular to the field in the United States. And so depending on this force order anisotropy, that determines whether there is a barrier between these two states at the switching field, or if there is a well, right, that guides the state in different ways. If there is a well, then the state just goes from a collinear state to a canted one in a continuous fashion. There is basically two second order phase transitions corresponding to the appearance of the perpendicular parameter and then disappearance of the parallel one. And then on the other hand, if the force order anisotropy has an opposite sign, then a barrier appears and the transition happens in the first order fashion. So joint magnetic effect in this material is due to the fact that really the anisotropy is of this type, right? And then within a narrow field range, the large polarization change is driven by the magnetic exchange friction. So you see that anisotropic interactions and the ability to compute exchange constants rather precisely is really crucial to correctly predict with physics. Okay, this was an encore in 2014. So then basically, let me say a few words just to motivate the discussion. So Anderson super exchange is basically the energy gain on the hopping that kinetic energy gain essentially on the hopping between the transition metal orbitals. Typical situation is when metal ions are bridged by the oxygen or can be halide ions or something like this. And then typical Hamiltonian that people can see there is a kind of Kanamori Hamiltonian that includes Haber term and the Kuhn circle couplings. And then there are hopings between the transition metal sites and oxygens. Then essentially in the ferromagnetic state for a single orbital model, Pauli principle includes hopping, but in anti ferromagnetic state hopping is possible and the energy gain is proportional to T square over U. On the other hand, ferromagnetic exchange is very rare and the ferromagnetic insulators are very difficult to find. Most of the ferromagnetic insulators are really ferrimagnetic. This is because the ferromagnetic exchange comes from the 90 degree bond where basically hopping happens to the orthogonal orbitals of the oxygen. And so essentially one can think of this in the following way. P orbitals are completely occupied on oxygen so it's P6 and then these electrons on the oxygen they can hop on the transition metal sites. When the two electrons of the same spin hop to this transition metal sites, then the two electrons of the opposite spin remain and this configuration is favored by the Kuhn circle coupling on oxygen. So because Kuhn circle coupling on oxygen is usually not very large that gives rise to the weakness of this contribution. So this is one of the typical mechanisms. So again I mean for a dimer spin one spins cannot hop and spin zero on a dimer up down minus down up. There is energy gain 40 square over U. And so then one can introduce with spin exchange operator computing the energy in the second order perturbation theory in T. So we are diagonalizing U in the Bayer Hamiltonian and then we are doing perturbation theory in T. Then as we hop the electrons from one side to another we come back to the situation when we exchange the electrons we come back to the situation where basically the wave function is the same up to a sign indeed. If the electrons if initial state was a triplet up down plus down up then by exchanging electrons you get up down plus down up again. So here you are coming back to the same state for a triplet but for a singlet you are coming back to the from up down minus down up you are coming back to minus that state because you change the spin and so minus sign gives rise to the minus here. So then that is encoded into this in this spin exchange operator. To S1 S2 plus one half that one can check is plus one for a triplet and minus one for a singlet. So this spin exchange operator keeps track of this sign essentially. And so then the energy second order perturbation energy turns out to be this way. Okay now the Lashinsky-Maria is also microscopically arising in a similar way but spin orbit coupling is involved in the simplest. So spin orbit coupling can be large both on transition metal ions but also on the ligands for example in iodide and bromine compounds. It was recently shown by work of several groups around them show that there are large anisotropic interactions due to strong spin orbit coupling on the ligands. And so basically one can sketch the process in the following way. You have two orbitals two perpendicular orbitals on the transition metal sides and usually exchange is going from yellow to yellow and then from second second electron goes from this yellow back to the to that one. In the Lashinsky-Maria process however spin orbit coupling assists the transition of the electron between yellow orbital and the green one. All right and then hoping happens from the yellow orbital to the green one on the other side. In the unbend bond in the symmetric configuration this green orbital is orthogonal to the yellow one so the hopping is zero. So that's why the Lashinsky-Maria normally happens when the bond is bent. And so it involves basically contributions of the spin orbit coupling right and then hoping between these wrong orbitals and then hoping of this electron here to the orbital on the left. So then basically the same story there is again this minus one factor encoded by spin exchange operator but there is also L dot s. And so then you compute the matrix elements of L dot s, L operator is minus i nabla right so between x, y, x, z, y, z real orbitals it's completely imaginary because of minus i nabla. All right so then when you add the Hermitian conjugate term this becomes a minus and then so you get essentially s1 2 s1 minus s1 s1 2. Then that gives rise to a trivial commutator commutator of spin exchange operator with the spin right s i s j equals e to the a b c s c and that gives rise to the vector product. So in this trivial way we can get very quickly the Lashinsky-Maria and so what we see here is that it's crucial that spin orbit coupling is strong that there is a hope into the wrong orbital facilitated by the bond bending and then you see that it's actually very important the distortion is very important right because the distortion of the bond controls directly with exchange energy so this d is controlled directly by the bond distortion. So normal understanding super exchange and also the Lashinsky-Maria and the symmetric anisotropy they are all strongly dependent on the orbital overlaps and the bond geometry. And so in the ultrafast experiments and also in multiferroics in physics of magnet electric coupling it's very important therefore to understand and be able to quantify the connection between parameters of the magnetic Hamiltonians such as d and j and the ionic displacements. So this coupling between spins and ionic displacements gives rise to spiromotive heroics particularly if you just can see the frustrated chain with the ferromagnetic nearest neighbor and the anti-ferromagnetic next nearest neighbor that gives rise to spiral states and so when I do the inversion let's say around this point then the spins don't change on the inversion so this spin goes here point in the same way so you see that spiral breaks inversion symmetry and thus gives rise to polarization in this case doesn't always have to but in this case it does. I mean it's possible also to break inversion and not in this polarization but in this case polarization arises its proportional direction is given by the spin rotation vector and the wave vector of the spiral and so again you see it's the basically inverse the Lashinsky-Maria so the coupling of spins to do the displacement of oxygen is what drives the polarization so once spins are non-colinear because of J1, J2 polarization arises to optimize this term. Okay then how to approach this problem so starting from 1987 Liechtenstein formula has been proposed in this paper by Liechtenstein Katsnelson and Kropov and Dubanov so they're adjusted to compute second-order correction to the energy of the magnet due to rotation of the two spins so they take two spins in the unit cell rotate one by some small angle another by another small angle it turns out that the energy correction is proportional to d5, 1, d5, 2 and then the second derivative is extracted from the basically a bubble-like diagram with the delta the kind of exchange operators in the vertices so then exchange can become computed this way with the Liechtenstein formula this has been implemented in a number of codes also using one-year basis so I think anomaly code it appeared around 2010 and then if I understand correctly then there was this paper that uses result A plus U more recently tb2j code implemented it and then there is also openmx that implements similar methodology and I also saw several talks at this conference that discussed this method then what would we like to do is basically to add the dependence of the Green's functions on the ionic displacements right so essentially here you have one over omega minus h and h the Hamiltonian through the dependence of tij depends on ionic displacements so essentially the formula for the exchange basically you just take with the delta g delta formula insert in the denominator one over h open plus h electron phonon and then you get essentially in the first order with diagram where the first upper and lower Green's functions can be corrected by the electron phonon coupling by interaction with this frozen phonon in the system right as ionic displacement so you get basically for me what g delta g delta h electron phonon and h electron phonon is conveniently implemented in epw where it's really implemented in the vanier basis so the electron phonon coupling in the vanier basis basically tells us the change of the open essentially between the orbitals zero and r due to the potential induced by displacement of the ion elsewhere right it can be one of these ions or ions somewhere else in the radius and so this basically gives directly this correlated correction another important part of the story is that if i write this simple super exchange formula there is dt over du times du and then there is also a variation of habert u or the gap right due to the shift of ion so far we haven't looked into this in detail and we use the fact that in centrosymmetric environment basically shifting ion to the left or right has to give rise to the same change of delta of u all right so that means that correction to u in a centrosymmetric environment at least is quadratic in the displacement so if the ion has is at the inversion center initially you don't have to consider that in the first order but that would be another very important point so this is a basically work of uh uh riot-olona so the point here is that magnet electric coupling is really uh it's coupling between spins and uh the ionic displacements comes from the dependence of the exchange matrix so there is basically s times the matrix of exchanges that depends on u in this following way on the ionic displacements to the first order and so optimizing so changing the ionic displacements leads to change of the exchange matrix and then spins rotate that leads to the change of magnetization okay so one particular case i would like to discuss uh of the use of this formula is type one multi ferrox so multi ferrox the materials with the existing electric and magnetic odors they have been studied for a long time and basically in type one multi ferrox uh for electricity and magnetism can exist but they are not induced by each other so they are weakly interacting and so for a long time uh so these materials are not considered as promising as a type two multi ferrox where magnetism induces for electricity on the other hand it turns out that uh in this materials type of one multi ferrox while there may not be a strong magnetic coupling in the bulk nevertheless at the main walls they may be a strong coupling let's consider a ferroelectric domain wall the amplitude uh versus the site index is shown in this plot so i have polarization downwards downwards and then polarization zero at the domain wall and then polarization changes to upwards this schematic situation so in a double well potential for polarization imagine some tetragonal crystal this uh uh corresponds to the four and so minimum in the left well left well then it's in the center at the domain wall and then on the right so polarization so the power distortion of the latest changes this way so it turns out that this uh change of this ferroelectric mode along the spatial coordinate also leads to the bond bending and changes in the tij and that leads to the modulation of exchange in the following way so at the ferroelectric domain wall exchange versus site index exchange changes slightly however the leads that affect the change of exchange leads to actually the dependence of the energy of magnetic domain wall on the coordinate right magnetic domain wall in this case turns out to be cheaper here at the ferroelectric one so that produces an attractive potential between the ferroelectric wall and the magnetic one now magnetic walls uh are usually rather wide and so pyro snobar barriers that pin them to the latest are often rather small so that is to say magnetic walls are very mobile which leads to the fact that it's probably easy to move the magnetic domain walls by moving ferroelectric ones even if there is no coupling no magnetic electric coupling in the bulk this effect change of exchanges at the ferroelectric wall we'll still try to convince the magnetic wall to move along and this seems to be uh consistent with the observation by Phoebe and uh at all uh in 2002 where they demonstrate the clamping of ferroelectric and anti-ferromagnetic walls so basically you take any structural wall can be very distorted for doing whatever uh ferroelastic anything and then uh if that modifies exchanges that will likely pin magnetic domain walls not necessarily ferromagnetic but also anti-ferromagnetic ones okay so in conclusion uh the exchange restriction uh can be concluated uh uh you know using electron phonon coupling and really it seems that uh it's just Liechtenstein formula plus electron phonon interactions and then uh one problem that is not uh resolved yet is the corrections to you due to the uh ionic shifts in the non-centrosymmetric environment and with that I thank you for your attention. Thank you Sager for very nice talk uh any questions from the audience so um maybe I can ask a question so so how do you deal so which uh functionals do you use for these magnetic states it looks like there are a lot of uh so it's mostly LD up SU or yeah right that's a good question so it turns out that uh actually reproducing the ground states even is not very trivial uh for uh nico to write that I talked about first uh with the LDA uh like PB is so plus you we cannot even get the correct ground state uh it turns out that PB zero works rather well at least I mean one can get to the correct ground state without any union of exchanges and then uh right I mean so the gap seems to be crucial like getting the right experimental gap seems to be very important because it controls the intermediate states that uh states that the hopping brings to so indeed uh exchange correlation is very important so do you think that's just specific for this system or would be generally true also or right I mean especially what's in an frustrated system where just for magnetic exchanges are there right that wouldn't be qualitatively hurting us but where there is a subtle interplay between different exchanges getting them wrong really changes the picture like uh makes it impossible to reproduce the characterized diagram so it's PB zero without so you suggest PB zero you don't add you right or you do both or right right PB zero opens the correct gap basically but you don't have to add you right no I mean uh with LDA plus you we couldn't get uh the same exchanges we tried different values of you okay even if we get the gap consistent with PB zero right the same gap is in PB zero with plus you we still don't get the same exchanges because the curvatures of the bands are different okay and uh yeah the TIJs are different so some subtle uh interplay okay thanks uh some other questions so so maybe I can ask another so this lift and send approach is that available in any code that I can just like download and use or is this yeah kind of like you know personal repositories no no uh so uh in principle uh so let's say tb2j implements it I think I will need also implemented just from scratch now uh I mean there is a bunch of codes that that does this and then we also have like a C version of it that uh we will basically write and to expand on it to get to the strong spin orbit coupling so I mean there there is a story where uh different so in the materials with strong spin orbit coupling like routinates and dyridates there can be a significant angle between the orbital and go momentum and spin and go momentum and basically you can kind of like in some strong some routinate I think it's like 40 degrees so then uh it makes sense to try to define like different exchanges uh like between l and s on different sides this was work of for we've been uh so yeah then but there is also a simple version for low spin orbit coupling that's also available in C and in Julia I mean we can share it we didn't share it I think openly because there are these other codes that basically do it so you think that tb2j is the kind of the most uh let's say user friendly for somebody who wants to get into this type of calculations tb2j yeah well right I mean also this amulet is basically working with quantum so okay okay we have a question over there just a curiosity because the dictation approach was initially uh suggested for uh I mean KKR approaches and now the the nice thing of the veneer approach is that you can somehow still get an atomic limit for and get in this way the jaj between two atomic sites but many functions are typically not super localized on just a given atom and in particularly if you linearize the same system by including more or less bands you might have that many functions are more or less extended how these are effects actually this the exchange coupling between between atoms right yeah so uh right we also did the tests with a different number of bands so just when you're rising a large window including more and more of the deeper levels of ligands and it turns out that the exchange is changing uh so it seemed that the window for like 30v uh below the Fermi level gives us exchange to like 0.1 millivolt but I mean there is energy denominator right that comes from the g-dota g-dota I think the question was more about empty states right like what if you change the no it is on the localization of the veneer functions in themselves yeah but if you add like deep states 30v below maybe that doesn't change that localization but if you add empty states maybe that will change it more so did you try to add more empty states or no no right so we do right right we do it uh the uh the empty and occupied states basically plus minus 30v oh so you did both yes okay thanks other questions no okay so let's uh thanks again thank you