 Yeah, okay, some power that can doesn't. Doesn't pop up. But yeah, it's all right. Maybe, maybe we'll, we'll do it. Okay. Yeah. Good. Okay. So, yeah, so. Okay, so let me. Okay, I will, I will introduce you so the, the next speaker is learning Levito from MIT, and he will talk about carol stoner magnet is in direct bounce. Okay. Yeah, thank you. Thank you. Thank you very much for the invitation and for giving me the opportunity to speak. And, you know, hello everyone. It's very nice to see you. Very remotely would be even better in person. And so the work I wanted to talk about is something recent. I mean, in Aaron terminology, it will be, it will be DC plus AC. And it's about how, you know, it originated from the interest in magnetism and in graphene bands, but I'll try to, you know, avoid as much as possible the specifics of graphene. So, you know, twisted or non twisted by Liz talk about it more generally question is how the band magnetism described by stoner instability is, is affected by Barry curvature and in, in the band, and, you know, whether very the curvature in the band has any, any effect on stoner magnetism at all. And if it, if it does, what, what is it the. So this page will summarize the main results and then, you know, the remainder of the talk will be will be driving it. So the main, main thing we find is that if, if there is a stoner instability leaving the band magnetization in the presence of Barry curvature there is a new interaction arising, which is the interaction between the orbital magnetization in the band with the very curvature. I don't see the pen. Do you see my pointer. No, we don't see. You do. I see, I see that. Okay, I'll, I'll, I'll try to use it as a proxy for the pen. Yeah, anyway, so there is there is this minus mp interaction arising where m is the orbital magnetization. And due to Barry curvature of the states which are filled by carriers in the band, and, and B is a pseudo magnetic field, which, which is proportional to that's partial as partial s of spin polarized electrons, and it's non zero for if magnetization is twisting. And then this quantity B is proportional to sine of it is proportional to chirality of that twisting texture. And, and so this interaction is, is there, you know, just allow it by symmetry, because it looks like you know general basic electromagnetic interaction between magnetic moment and external magnetic field. But we'll, so here both M and B play an unusual role. B is not an external field that is the intrinsic field arising to you to twisting of spin texture and M, M is the orbital magnetization to the very curvature. There is, and this interaction, you know, we'll derive it, but maybe right now, we can say that the effects that will have can be understood by the cause effects relations so in the usual, in the usual magnetism when we have magnetic moment couple to be with you be as a cause and M, M was the effect so we apply magnetic field and magnetic field polarizes, polarizes our magnetic moment here it's the reverse of that. The orbital magnetization is present in the band with very curvature from the start. And then, if it's present it, it couples linearly to, to that to the chirality density which we'd be proportional to, and because of that linear coupling is to induce chirality. I mean there are other terms in the energy proportional to be square and we'll talk about it later, but, but this linear coupling will tend to generate chirality and that's the main, that's the main idea so the goal is to derive this interaction and discuss the, the effects of this interaction and, and, and this interaction as I already mentioned, it's allowed by symmetry. And so it has to be there, you know, just for general reasons. Interestingly, this is an interaction that you can see between the spin quantity, which is B, and the orbital quantity which is M arising without, without any spin orbit interaction so it's an interesting case of a, what we can maybe call synthetic spin orbit, spin orbit interaction, right? And yeah, so it, so this interaction will favor spin textures with non-zero chirality and some chiral order that can be either a square-mian lattice or a square-mian liquid or something, something else. So if this interaction is winning over other interactions it will, it will generate some interesting, interesting chiral states and then on the lower right over here is a schematic, schematic then structure that, you know, a very caricature presentation of a, of a band of graphene bilayer with gap and just by transverse fields similar to what you heard in, in many talks this week and previous week and I will be using it just, you know, as a, as a caricature for what we're interested in. So we'll, we'll consider carriers and one with band and, and they're, as they abandon chism and then how a very caricature affects that. And, and there'll be two phases, there'll be a phase which is like a conventional stoner phase when the station is uniform. And then also a phase with magnetization twist and their competition will depend on the details, the details of the interaction. So that, that's, that's the, that's a quick summary and then now let me just check which time was left, yeah, okay, so I can, I can talk about how this will be right, some links. And, and also here is the paper on, on, on that, on that story that you can find an archive, the work done with Jiu-Ju Dong, which is an outstanding, outstanding young man finishing PhD next year. Right. Okay, so good. So let's now, let's now maybe dive into, into, into, into some details. So what I want to consider to, to derive this carillon traction is a two-band, two-band model, something like this Dirac, Dirac-Bent model for curfew and bilayer and one of its, one of its valleys. And, you know, G is the gap-primary changes by an external electric field for a, for a particular plane. And G has new effects that, first it creates, I think I had a better picture with Jiu's show here. This is what G is, right. So G has new effects. And it opens the gap. B, it flattens, flattens the bottom of the conduction band and of the balance band and makes the band very flattened that, that makes conditions for stoner instability more favorable, easier, easier to fulfill. And indeed, under thin bilayer over past two years, many groups reported, reported magnetism, orbital magnetism of stoner type with different kinds of, several different pages of different polarizations. And yeah, so it seems to be a system to, to, to, to, to build upon. Also, it has very curved chest because everything we need. Good. So I'll take that for kinetic energy and then add exchange interaction to it with some, with some potential you. So here, you know, I am assuming that somehow started with microscopic interaction and then was great integrated out the irrelevant degrees of freedom and extracted the exchange interaction that, that dominates the, the, the, the magnetism. So this is, of course, not a my fully microscopic model, but a simplified, simplified model, a toy model, but that's what I'll be watching. But anyway, so if we want to, if we start with that, then the step one is straightforward and I'm not going to talk about them. Great detail just, we can just do ordinary in the field, mean field calculation for stoner magnetism and three groups in three groups in fields by Herbert Stratanovich and a little subtle point. And that's what we'll do. However, we'll allow popularization spin polarization to have spatial dependence. So we're not going to restrict ourselves to the utilization. And so in this, in this calculation, basically we'll be doing this, what's shown on a lower right will be considering, you know, bent, bent polarization in one of the valleys, another valley, both valleys with simplicity. It's easier to think about one valley at first and then discuss what happens if we have both valleys. So we'll, you know, we'll consider, we'll consider, sorry, my animation. Yeah, no, it works. Yeah, so this is the spin polarization transition will be interested in, and plus and minus by the way mark the sign of very curvature and in the particle band K and K prime and the whole band K and K prime. Yeah, each other as they should. Right. So, yeah, so like I said, we start with this Hamiltonian. The first thing we do is, you know, the couple spin spin interaction by Herbert Stratanovich introduce ordering field age, and both agents in polarization are allowed to vary in space. And the interaction. I mean, there are no require specific requirements on interaction but just for example, I will say that it's some interaction with the with the interaction radius psi and will be will be using it later. So, yeah, so then the strategy as always is to do habit Stratanovich and then include the ordering field the same on term couple for spin couple to ordering field in the single particle Hamiltonian and the great out for me on the saddle point that I'm not going to talk about it in great detail. There is a paper cited on the lower right that that discusses it. And that that's a paper from from the DC DC period in Aaron's nomenclature, maybe one thing that will be useful for for for the next page is that if I do this decoupling and then take, take that a square interaction. For the ordering field. It has some non locality because of because of K dependence of the interaction you and if I expand on K. Namely expanded gradients I can from from this from this term I can extract spin stiffness for my for my clinic problem. And then that value in stiffness will will matter because compete with my curve interaction later so let me introduce it here and just say that it's a standard standard procedure that we always use when we do. Okay, good. Tell me if there any questions I don't see chats, but maybe good. Maybe not good but whatever it. Good. So, right, so then, I mean, the next thing to do is to integrate out for me on with with a free particle Hamiltonian with in which ordering field age is arbitrary position dependent field. We get this free energy which is, which is shown on top and then, and then it's written so that there is no preferred spin access in conversation access. So it's fully covariant and and then next thing we can do is, you know, also standard maybe a little less standard, we can, we can perform a local spin rotation. To align spins with, you know, with with with the z axis and when we do that, when we do that from the spin rotation matrix, a spin dependent. vector potential generated. So if you do this, if you do the spin rotation. And then, then this free particle Hamiltonian is altered by 50p by some vector potential and a will be spin dependent and proportional to the gradient of rotation, rotation angle angle, and so that that's so this is this is what this is what looks like. So now ordering field is aligned with z axis, but, but, but the free particle Hamiltonian couples to couples to a vector potential and this vector potential in general will be non a billion so it will depend on. Yeah, I don't have a pen but instead of what's what's here a sigma three that would be a one sigma one a two sigma two a three sigma three. And, and that so in this way it's you know completely general and no approximations made. However, since we assume that, that local pictures that will be some, some uniform locally uniform simplification with orientation slowly varying in space, it will, you know, we will, we can make a, you know, a diabetic approximation of ignoring ignoring the components in that vector potential that couple up spin and down spin down spin bands and if that that's possible with the diabetic condition of school field if our spin spin polarization is very insufficiently slowly and that's a condition for that and if we do that, then our, our vector potential becomes a billion or approximately a billion so we're going to work on this approximation. And then, yeah, then we can take, take the magnetic field corresponding to that potential it's, it's curl and and discover that that that that curl is actually equal to the spin kerala density that that we introduced earlier as partially as partial as it is also equal to the topological density if you I mean if you think about spin. Spin defining a mapping of a two dimensional plane on to a unit sphere, then the degree of that mapping would be an integral of that quantity scale by scale by divided by pi. And it will give a number of turns a number of wrappings, the spin configuration makes makes of the, of the unit sphere. And so, so in this way the vector potential that we generated in this way by spin rotation can be related to a magnetic field and be the topological density associated with with with kerala. Okay, so far so good. And, and so then, I mean, next thing we maybe before we proceed, let me just, you know, mention a bit history. So, so these, these eight years that out of electron electron interactions in in a magnetic systems and interaction and the emergent gatefield can arise that corresponds to spin twisting. They are of course, not, not new and they've been popular very popular in the early days of high temperature superconductivity and start all started with, with the Anderson idea of RB RB correlated who prays and then that was reinterpreted as a gatefield in a series in a series of papers and some elements of what I will be talking about similar to what the what these papers did. However, the mechanism they considered didn't involve very curvature and so the outcome. The outcome was quite different from what, from what I will be discussing, however, on some on a formal level, there are some similarities, especially with this paper by Herbert Schultz 1990. And then, yeah, and, and then maybe going further in history, then later, about 10 years later, maybe 15 years later, people. There wasn't a revived interest in this topic of emergent gatefield in connection to non collinear in connection to how conduction electrons behave in the presence of a non collinear background. There have been many examples found out where non collinear magnetization induces a gatefield arising because the electron spin wants to track local spin and if local spin due to magnetic, you know, non collinear magnetic is twisting then the electron spin will be twisting and that will generate that will generate very curvature and the very curvature will impact the dynamics of electrons in the same way as some magnetic field applied to them. And this will give rise to to to an interesting contribution to the to the whole effect, known as the logical whole effect. And if, if, if that if the spin texture has chiralcy that would that would give rise to those chromions to land all of that, all of that has been has been in a literature in connection to spin textures and magnetic materials and how how conduction lecture and see them and there is a nicer sighted here the bottom by McGawas and the aura of the south city, but I mean this literature is very, very light. Anyway, so, so, so some of these eight years have appeared in some form I mean the connection to very curved chasing because you and actually very curvature is what generates chiral interaction that allows to somehow realize some of the, some of the ideas that were forwarded in this. Sorry, there is one question in the chat. It's the asking whether the field is homogeneous in a space. Oh be is proportional. Yeah, the field be the magnetic field is proportional to as partial as partial as so whatever, whatever spin density will do be will replicate that so it may may be homogeneous or may not be homogeneous. I will tend to assume I mean using this a diabetic condition. I will tend to assume that the spin magnetization varies but very slowly so everybody locally one can one can make an approximation with nearly uniform nearly uniform spin. There is no specific density and therefore nearly uniform magnetic magnetic field, but no special assumptions. Otherwise, we'll get to that I mean, when we'll do actually in next page we'll get to that. Good. So, so now that we derived the effects of action. I mean, this is our effective action. The trace log term is what happens if we integrate integrate over fermions. And then, and then this term on the left on the right is the spin stiffness term that arising standard in a standard manner. So it's partial as squared. And, and then now we can take that and ask, and, you know, try to settle point before we look for settle point let's, let's, you know, let's assume that even if the spin dependent vector potential is there and zero and chirality is there and zero, it's weak. So, you know, it's slowly, slowly twisting states. So, so I'm allowed them to do some kind of expansion and powers in powers away. Now if I do expansion powers away because of gauging variance. I'm only going to get terms proportional to gradient today. And, and so that these, if one carries out this expansion. The important thing is that there are terms which are first order, first order in in be the acceptable and also in a They are, they are allowed by symmetry and they would, you know, what you get a few, if you do the expansion and at linear order in a the term proportional to curl away. The first power of curly way comes out multiplied by multiplied by very curvature. And then also, you know, say, cast. And also, a term second order in the comes out and probably turn the pilot and if I'm, if I'm thinking about, if I'm thinking about small, small a expansion, I think we can limit ourselves to what you see here. And yeah, and so the zero order term is the ordinary stoner stoner carrier, carrier energy, which is written here the kinetic energy in the spins of the dance type plus plus interaction term that described energy was when you gain polarization. And then these two terms are the interesting ones. So delta amp is the orbital magnetization of carriers with up spin minus orbitalization of carriers was with down spin. And, and then this, and it's, it's multiplied by be the be the chiral density. The, the dynamic energy is simply the Landau diamagnetism in that, in that magnetic field so chi chi is simply equal to, to land out a magnetic stability it's a positive number that depends on depends on then, then this version. Right. And good. So, and, you know, this is a, you know, this is a, this is the final answer for that, for that chiral interaction. And, yeah, so let me maybe emphasize again so the interesting thing about this interaction is that it's an interaction between orbital, orbital degree of freedom with you to do the orbital organization in, in, in, in, in a band with very curvature and spin degrees of freedom that go in the, in the chiral density, and it's arising without spin orbit and traction and other interesting features of this interaction is that it is as you do invariance so it's very different from, from ordinaries in orbit and traction if you imagine spin orbit attraction let's say in an atom, then that would look like L dot S where L is orbital, orbital momentum and as a spin momentum, and now if I perform the rotation that can only be invariant if simultaneously with spin rotation, I perform rotation of the orbital rotation, I have to perform rotation in real space and spin space by along about the same axis by same angle, otherwise the interaction won't be invariant. And that's what we usually, that's how we usually think about spin orbit attraction because it locks locks spin variable orbital variable here, however, this interaction is as you to invariant time the rotation in spin space is no, no, was nothing happening to orbital degrees for you. So it's as you to invariant alone without, without orbital degree of freedom as it should be because there is no spin orbit and direction in a multi on start with, right. Good. So, so that's, it's more or less the, the, the essence of it and then we can talk about about the instability towards higher order. So, to understand this instability have to consider the competition between all these spin, spin dependent terms that we derived the stiffness part and the, and the orbital and the chiral interaction part and that report. So this analysis can be carried out in a, in a fairly straightforward way, using some knowledge of what, or what we know, what we know from a literature on instant times we can actually try and relate, relate the spin stiffness energy to chirality to chiral interaction, using some standard identities from, from skirmjohn literature I think they are not the only equations. That's it equal, equal to completing, completing a square. So basically the way it works is that you can, you can take a square gradient of s plus minus with as partial with s cross partial s square it and discover that exactly after you open up a square, the partial square term generates, generates this and the s partial s square also generates the same, same term of the same form and therefore one half, one half disappears. But the cross term of these two terms, generates the topological density and then from that, from that you can infer two things first of all you can infer, infer that the spin stiffness, spin stiffness energy is always greater than the absolute value of the, of the topological spin density absolute value. And, and also that greater or equal and then also that the energy is at its lowest when, sorry, the energy is at its lowest when, when this is not greater or equal but equal, equal side. And so if you take spin configuration, spin, spin texture such that this is an equal sign and these configurations exist and one can, one can work them out and write them, write them explicitly they're, they're the, they're well known in the instant literature, then, then this, then the stiffness energy basically becomes proportional to be the topological density absolute value. Right, and so then our energy is expressed solely, solely through topological density like in this formula in the, in the red box. And now we can, now we can analyze the instability and so you see that. You see that as the, the chiral interaction, they was nonzero B, if there was no term with the absolute value then it would, it would want chirality to be nonzero if, and then if J the stiffness is sufficiently small it will win over stiffness energy and will, will produce a chiral state, and then the equilibrium value you know, settle point value of B will be found by competition of the terms linear in terms quadratic can be and as you can see here at solutions be zero, be equal zero and stiffness is big, or orbital organization is small and be which is nonzero and that that's when stiffness is small, or orbital organization is large. Right, so therefore we expect that there'll be two phases and this is the condition for, this is the condition for the chiral order to win. And so if you take this, take, take, take this a step further and actually take the energies of these different, you know, have to add, have to add the energy of a spin polarized phase found by stone analysis. Get something like this phase diagram, where, so let me say again with an axis, the lower axis is carrier density so it basically tells you how much, how many carriers you have in one of these conduction bands on the lower right. The lower axis is the displacement parameter that's likely shown here. That's a parameter that opens the band gap. And then, I mean, you can do, can do the numbers and you see that and you find that there are two different ordered phases one, one is similar to what you know everyone analyzed already for graphene bilayer and tri-layer in which the organization is uniform. And then there is a new phase arising at lower carrier densities, arising at lower carrier densities in which magnetization forms a twisting texture with some definite chirality. The transitions here, I mean, in this calculation the transitions happen to be first order. I don't think there is anything fundamental about this being first order. Some of these transitions would be allowed by symmetry to be second order, but that doesn't happen because of how it works out in the dimensional system. Another thing to mention is that as you can see here, you know, something interesting going on, this chiral phase when it arises, it actually lowers the threshold for stone instability. So without chiral, if you wouldn't think about chiral phase, the phase boundary between unpolarized phase and spin polarized phase would be a straight line that runs along the diagonal. And then the chiral order pushes the phase boundary down meaning that it is easier to reach. You need weaker interaction, less stringent conditions to achieve stone instability. This can be understood in terms of, you know, this linear dependence, I mean, goes back to what in Russian literature known as Larkin-Pikin, Larkin-Pikin effect that if you have a phase transition and then you have some other fields which couples linearly to the order parameter then that linear coupling A makes transition more accessible and B turns a second order transition into a first order transition. And then there have been many, many examples worked out in the correlated electron systems and particularly in the work by Premi Chandran, Pierce Coleman, who considered a very interesting analog of that same phenomenon in electron nematic phases, but I mean the general phenomenon is that as soon as you get, once you get a linear coupling to some external field and that external field has its own dynamics, it will A lower the threshold for the instability and D will, you know, turn the transition from second order to first order. Yeah, so that I think this is more or less all I had to say. Let me just, oh yeah, now if we, I mean, if we go in that chiral order phase and ask what do you expect to see there, I mean, I think that this is quite well understood. And this is, this is one of these phase phases where we will have skirmjons of, you know, particular chirality and they, you know, they, as a reminder for everyone skirmjons are topologically protected configurations of twisting magnetization with a number being the integral of topological density and logical density is up to a factor of two pi is the same as my is the magnetic field and they can form different phases. I mean, they usually form in various systems where they appeared quantum whole systems and they and the layered magnetic systems they tend to form periodic letters as they form to tend to form skirmjons crystals. However, and there are many, many of these that have been predicted and observed. However, quantum fluctuations are strong or thermal fluctuations are strong. Then, then we can have a situation where we have a chromium liquid in which, in which the long range spin order spin order is lost. But, but the symmetry associated with picking a particular chirality takes a definite value so it will be the skirmjons liquid with longer spin order destroyed but chirality being present will, will be an example of the symmetry breaking. And again, there are many, many examples similar behavior in the literature including the paper by, by Pearson, Bremy, as they just cited. Okay, so maybe, maybe let's stop and see if there are any questions before I conclude. Okay, so thanks. Is there first any questions. Thanks a lot for the talk. Are there. So are there questions in the room. Yeah. Hi, when you wrote the Hamiltonian first in the single particle Hamiltonian there was no spin index. So, is it. So how does this, how does the, the single part. Is that intentional. Yeah, so the single particle Hamiltonian was kind of energy only it was spin independent. What I meant is that it's identical for all spin species. So, let's say in Griffin by the way we have, we have or is you the spin components you can call them there is there is an ordinary spin spin one half variable. And there is a value degree of freedom and they, all of them have the same, have the same band dispersions. So they, they would all be described by by that Hamiltonian that I wrote so if we, if we focus on one, let's say in one of these graphene bands and, and consider the spin polarization in that band that's that will be the simplest example but yes they there is no explicit dependence on spin in the original Hamiltonian because there is no spin orbit. And the both that k k k valley and k prime valley has opposite very curvature right so that's right. Yeah. So does that mean, like, when it's found this kind of order it's spontaneously break the valley symmetry or doesn't have to be. Yes, so if I have, if I have, if I have magnetic instability in both k and k prime. Well, a, these magnetic constabilities are at some crude approximation, they, they decouple because it's changing traction that drives drives and stability is mostly intra valley and not into valley. And that's the reason for the reason that can he prime value is a very far away in case space compared to the size of the fact of 100 that that allows to treat each valley separately, the magnitude of each valley separately from the other valley. Now, if, if everything is very symmetric, then there will be a magnetic state and valley k with one hierarchy. And then we'll have a situation when we, when we have two of these states that they talked about occurring occurring simultaneously one and K one and K prime. However, there is something that breaks valley symmetry. And it can be either an external magnetic field or in twisted by layer, let's say in twisted by layer graphing as I'm sure people, people talk to repeatedly during these two weeks that that happens spontaneously so you have spontaneous valley symmetry breaking then that just won't happen in one valley, but not, not in the other valley and then, then that that will keep the same value. Am I answering the right question. Yeah, yeah. Yeah, that's that's answer. Yeah. Thanks. Thank you. Very nice talk. I have a question about the fluctuations. That could possibly break the spin order. So the term of the spin or be term that you consider is basically quadrack in momentum and cubic in spin that breaks time reversal symmetry. And if situations are strong what kind of gauge dynamics, would you expect. So for example, if into the magnets if one considers in some sometimes if one considers a carality term in the Hamiltonian that can break the spin order. Can break time reversal leading to some carol spin liquid. I think they're having coffee. Sorry. Yeah. With you on emerging gauge field I think you mentioned big month of mayor paper with a similar situation. Now in your case what kind of gauge dynamics and what kind of field theory would you expect to be too early to tell I mean this is something we're working on. I mean, I, I, I think it will probably be similar more like what Nagao San Lee talked about, but maybe too early to tell I don't, I don't want to get it. I don't want to get into that. Other questions in the room. I have a question in the, in the chat. They ask in the term M times be why M is the very curvature orbital magnetization. They ask whether it is derived, whether it's derived. Oh, it is totally derived. I mean, it's, I mean, we. So if you look in that paper that they say that it is derived there and you can also argue by, by, by symmetry that if you have, if you have magnetic, if you have magnetic moment of some orbital magnetic moment of some or any, any magnetic moment of some kind. And if you have some magnetic fields, you the magnetic or real magnetic couples those magnetic field and then the coupling should be of minus mb mb form. So that much you can, you can anticipate just, you know, based on general grounds on symmetry but yeah we went through all the details of the derivation and it's pretty, pretty straightforward. I mean, what I talked about is the less straightforward part and then if you, if you actually, then consider, consider, consider the magnetic fields, consider Dirac electrons in the presence of the field and then then you can, you can derive the orbital magnetization and be that this orbital nation, which by the way is exactly for the standard form that chin chin new dirac, you know, introduced first some 15 years ago, and you know we, we agree we agree with them with them completely on the orbital magnetization, which is simply the, the very curvature of all the field states integrated over that gives the magnetic moment that couples to magnetic field in the way talked about yeah so it's derived. It's, it's on the paper. So, thank you. I think there are no more questions. So we thanks Leonid again. And so now we go for the coffee break and we are back at 415.