 Hello everyone, so We start today with the first lecture by a professor of the rebound with Can you hear us at if people connected that through zoom maybe just write something in the chat? Yes, okay, fantastic. I think you can start I'm not probably not going to be watching the chat Okay, so Again, maybe for those who are connected you can you can wait maybe until the end of the lecture to ask questions You can either write them in the chat or you can raise your hand virtually and zoom and then we will We will unmute you and you will be allowed to ask your questions at the end Yeah, I don't mean I don't mind at all being interrupted So, you know if you ask questions in the chat or something somebody somebody will alert me and then Read me the question or something Uh, good morning. It's wonderful to be here. I'm a longtime visitor of ICTP. I don't It's not quite an uncountable number of times that I've been here, but it's wonderful to be back for the first time after COVID Welcome everybody So what I've decided to do to be a little bit orthogonal to what was done yesterday is to focus in on a couple of topics about use of 1a functions one is for a derivation of the theory of orbital magnetization and the second is about something called hybrid 1a functions And in the context of that I'll talk a little bit about topology and topological obstruction So here's the overall outline of the talk On the first part as I said will be about theory of orbital magnetization With a little bit about results just to add a little bit of something other than methods and theory in here and then And then about hybrid 1a functions towards the end and I'll see how I do with time The the top row here, of course are the people who wrote the reviews of modern physics with me. This was a wonderful Activity we were those were in the early days of Skype we had Skype meetings with all five of us from from multiple continents and Nicola we you know we published the paper in alphabetical order of the authors But luckily Nicola was the first in the alphabet because he was the obvious first among equals in this and I enjoyed his historical presentation very much yesterday the only thing I would say is that He makes it all sound so seamless and direct in retrospect at the time I remember going down a lot of blind alleys and Trying the different things that didn't work and so on so don't get the impression that everything happens right the first time And the same was true of orbital magnetization as I'll come to in a moment The initial papers with were with Timo Tonhauser and David it's only and with Raffaella And then there's a later paper with Evo and myself I'll give you the references later and then the people at the bottom here are others Mostly students of mine or collaborators who have been involved in the work at various times And I'm not going to be careful about giving proper acknowledgment as I go along in a few cases. I will but but not always Okay, so let me start just by setting up the problem of orbital magnetization And what is the problem of orbital magnetization? Well, it's similar to the problem of electric polarization in electric polarization real crystals have some Distributed charge density There's no way of dividing the unit cells into discrete Units with vacuum in between unlike what's shown in the textbook at right and therefore What we conclusion we come to is that even if you knew exactly what the crystal in charge density was Since it's only related to the polarization by a divergence of the polarization And if I add any constant vector to the polarization Uniform through space. I'll get the same charge density So that tells me that the charge density does not determine the average value of polarization And so what we learned we as a community learned Back in the 90s was that charge density by itself is insufficient even in principle to determine polarization And the same thing is true of orbital magnetization So if you imagine that you know the orbital current j of r Everywhere locally inside a unit cell that's not enough to determine the magnetization because the The the current is only the curl which is a derivative of the magnetization So if I add a constant magnetization I don't change the the current and again if The world is made up of little circulating currents with vacuum in between then we can make a definition based on unit cells But in general we can't and so there's a problem with orbital magnetization Now the problem with orbital magnetization is actually a much milder problem because in actuality Real orbital currents do look a lot like the picture on the right-hand side There's currents are mostly in the d-shell or f-shell of a magnetic ion and there's not much current in between And so for many years people have had and continue to calculate orbital moments by integrating the currents inside some spheres and Assuming that the current is zero outside those spheres and that's actually a pretty good physical approximation As we'll see later But it doesn't answer the question of in principle. What is really the correct way of computing the magnetic field? Sorry the magnetization and that's the problem that we set ourselves a long time ago When we did this work So let me start by again by analogy with a Toy derivation of electric polarization. This is not the derivation that I really prefer because it doesn't Discuss orbitals. It doesn't discuss adiabatic currents, which I think is the proper way of doing it But here's the here's the here's the idea. Suppose I have some let's imagine a two-dimensional system I make a sample which is maybe a hundred by a hundred square unit cells So it's a finite sample these size are the extended eigenstates of the entire sample a little bit like block states But but with boundary conditions and from those from the set of all whatever 10,000 extended states I form maximally localized orbitals Which is the the finite system? Analog of one a functions and those are what I call WM and then each one of those is Exponentially localized and I sum the expectation values of the position operators of all of them And that's the dipole moment of the entire sample And then I divide that by the sample area to get the polarization and so that's If I do that in the thermodynamic limit, I'm dominated by the localized orbitals that are in the interior of the sample and those look just like bulk one a functions So all I have to do is take one one a function. I'm assuming here just a single A single band insulator. I take the one one a function localized in the home unit cell That's what this zero in the cat is a notation for the one a function in the home unit cell I take the expectation value of X and that gives me the polarization So that turns out to be the correct answer, of course There's a little bit more algebra to put it into the form of a berry phase As you heard yesterday you take the cell periodic part of the block function you put it into this definition of The berry connection and if you integrate the berry connection over the B1 zone you get the polarization You can also take an additional wave vector derivative of the berry connection to get a berry curvature and the berry curvature is responsible for the anomalous Hall conductivity Okay, so And here's the derivation again that it shows that if I Take the X expectation value of the position operator it gets related to The berry phase because when you multiply by X and you do an integration by parts you get a d by dk Which appears down here and so you get this berry phase expression for the polarization So I'm sorry. I did that a little quickly because I just wanted to set up an analogous Pathway for trying to derive orbital magnetization, which is what we set out to do with with these collaborators On the rest of this slide so the highlighted green thing is what I'm going to talk about in the next 20 minutes Just before our work there was a semi-classical derivation by the group of Chen Yu At the University of Texas from a very different point of view from the point of view of wave packets and the classical evolution of quantum wave packets and The results of the two theories turn out to be the same Although the semi-classical derivation is a little bit more general because it also works for metals And then we had a second paper where we dealt with a multi-band case There was another derivation again from the Chen Yu group based on a long-wave derivation a little bit later and Then I'll talk a little bit about the multi-band and magnetic circular diachryism paper with Evo Sousa So these are the relevant references for for those who want to delve into the into the theory so for for the next 20 minutes my derivation is going to be based on a Two-dimensional ferromagnetic insulator with the magnetization normal to the plane It's a single particle Hamiltonian. There's no macroscopic B field Just a single occupied band and spinless electrons And so it would be something like the Haldane model is a is a model of spinless electrons But it has complex inter neighbor hoppings in order to break time reversal symmetry In order to get a non-zero orbital magnetization or you could imagine some continuous system in two dimensions that would have a single occupied band Okay, so I'm going to try to follow the same logic that I followed for The electric polarization. I write down the orbital moment of the entire sample Let's say I take a hundred by a hundred a finite sample. I calculate all of the extended orbitals I compute the Circulation x times v y minus y times v x is basically the angular momentum And then I multiply by q and divide by c to turn that into to see to turn that into an orbital moment And I use the fact that the velocity It's the commutator of the position with the Hamiltonian and you do a little bit of algebra and you get this This formula at the bottom for the orbital moment of the sample as a whole that's an exact formula and then you divide by the area of the sample and That should give you the something that has the units of orbital magnetization and If you again assume that this quantity is going to be dominated by the 1a functions in the center of the sample Since the ones at the edge are just a set of measure zero relative to the to the bulk then you would say that the macroscopic magnetization ought to be just given by Evaluating this matrix element of x times the Hamiltonian times y for the 1a function in the home unit cell divide by the unit cell area and Since this involves a kind of a circulation we call this the local circulation orbital magnetization And so our hypothesis initially was that that should be the correct definition of orbital magnetization and And if you turn that into a case-based formula using the same kind of tricks that were used for the electric polarization You get you get a formula that looks like this. So again, this is written in two dimensions It's an integral over the two-dimensional Brie-Wen zone And then what you have here is something that let me go back just to show you this looks a very much like the Berry curvature So where did I have a Berry curvature? Well, I didn't write it out. Did I okay? But but basically if the Hamiltonian were not in that matrix element If HK was not here, this would be the Berry curvature, but instead it's got the Hamiltonian mixed in in In the derivatives of the wave wave functions with respect to wave vector Okay, so we tested this on the Hall-Dane model Hall-Dane model Is famously a model that has a topological phase, but we were studying it in the non-topological the trivial part of the phase diagram it has some complex hoppings that make the orbital magnetization be non-zero and within tight binding you can calculate the current flowing on every Every bond and what you're seeing there is a lot of little arrows that show the currents that are flowing on various bonds and And so you can map out the current everywhere and you can calculate the total magnetic moment of the entire sample and divide by the sample area and I Guess that's what the red curve is and then you can also calculate the What we thought was going to be the magnetization the local circulation from the formula that I gave you based on the one a function and we did that and you know the results just didn't agree at all and Raffaella, I'm sure you remember We kept kind of bugging these two postdocs to go back and check You can't be right. You must be making a mistake somewhere but they kept coming back and saying no no we don't think there's anything wrong and eventually we realized so this is this confusion so eventually we realized that There really was something wrong and it wasn't wrong in the calculation it was wrong in our in our theory and So this is a lesson really that sometimes trying to implement something on the computer even in a simple tight binding model teaches you Something you can check whether your theory is sensible and in this case It told us that our theory was was missing something and and here's what it was missing So what we really need to do is sum over all these localized orbitals of the the circulation R cross V and I can rewrite that as r minus r bar So I really should have labeled this as our bar subscript s. So that's the center of this particular one a function Plus our bar cross the velocity matrix element So all I've done is add and subtract this our bar term but the local circulation piece that we calculated is really this this first one labeled in green and We did not include the second one Labeled in blue and and why not well So the idea is this if this one a function has a non-zero velocity expectation value Then our bar cross the velocity is some orbital moment However in the bulk of the material The velocity matrix element of the one a function has to be zero Because a bulk one a function if the if the velocity matrix element was non-zero It would mean that in the ground state of the crystal there's a non-zero current flowing and that's not allowed to happen And so it must be that for the bulk one a function the expectation value of the velocity operator is zero And so there is nothing coming from bulk one a functions But what about those one a functions by which I really mean the localized orbitals? At the surface is it possible that they have a contribution and what we found is that when we looked instead of the currents On the various links instead we evaluated the matrix element of the velocity operator in each one a function We saw a picture that looks like this So there's a one a function localized basically on each one of these blue dots There actually is a very small arrow on the next blue dot in from the from the border But it was too small to draw but basically what you see is that there's a current it looks like a current that's flowing around the perimeter and It does not vanish in the in the extended limit Obviously because if I make the sample twice as large I still have the same current flowing around the boundary and that corresponds to the same Contribution of orbital magnetization and so we called this the itinerant circulation In our model calculation. We saw explicitly that it did exist and we calculated it And then added that to the Local circulation and finally got agreement with the direct calculation of the orbital moment of the entire sample so this was a It was a fun project actually because it was interesting to Be brought up short and realized that we had not understood something and were forced to understand it better Okay, so there's a few steps I of derivation here, which I've left out Which it looks like in this picture This itinerant circulation depends upon the properties of the one a functions at the boundary But there's an argument and I if I have time at the end I'll go through the argument I have a few slides at the end. I can pull it back up Which shows you that? in the end you can rewrite that itinerant circulation contribution as a As something that only involves bulk one a functions again You start by plugging in the velocity in here as a commutator of the coordinate operator and the Hamiltonian operator But there's some additional Discussion about how to get the boundary contribution to look like a bulk contribution But at the end of the day the itinerant circulation is written this way. This is the Some of our lattice vectors since I only have a single band The one a function label is the same as the unit cell label. So Rx is really Labeling a one a function and that's the x-coordinate of the lattice vector where it's located then I calculate the y expectation value of the matrix element of between one a functions in the home cell and cell are and Then I go back again and calculate the Hamiltonian matrix from h from r back to zero and then the other term And if you convert this into k space again, the algebra is going a little bit fast here, but it's a similar trick Where when you have an x operator or a y operator you do an integration by parts So it turns into a d by dx or a d by dy which really are the Berry connections and Hamiltonian operator of course Can be written in terms of a Fourier transform of that a band energy and when you put all of this together so you basically got a the Antisymmetric thing turns into a curl of the Berry Connection which is the Berry curvature and you get this very simple formula down here so the itinerant circulation is Just the Berry curvature at a particular point in k space Times the band energy at the same point in k space integrated over two dimensions So if I didn't have this energy e of k here, this would be a basically the anomalous Hall conductivity But with the energy in here, it's the itinerant circulation contribution to the orbital magnetization Okay, so here's what the final theory looks like all written in k space of the derivation was done in Terms of one a functions, but now it's been converted back into k space where in some ways It's easier to evaluate actually as will I'll come to in a moment Probably the best way to evaluate it is using one a interpolation But in any case one can go back and forth between the the one a basis and the k space basis In a straightforward way So this derivation works only for Insulators because I need to have one a functions to find and for that I need filled bands throughout the Brewan zone and as I presented it it was only for single-band insulators and so a More general derivation has to include doing everything in three dimensions doing it for the multi multi band case Including spin orbit turns out not to be particularly Subtle it's it's rather straightforward including non-colonial spins and so on that the But it's still for insulators if you want to go beyond insulators then the derivation of the Chen you group using Semi-classical argument based on wave packets Gives you the proper expression for a metal and in that case There's a Fermi energy that comes into this expression well It's hard to see exactly what the relation is between these two expressions, but it turns out that they're identical for insulators and the The bottom formula is the proper It's the it's the proper extension for for metals So that's something that's unlike the electric polarization problem is only defined for insulators and not it's not even a problem for metals In the case of orbital magnetization of course exists for both So here I want to give a call out to the paper with Evo that came a little bit later About the relation to magnetic circular dichroism and really in this paper. We also did a more careful job of treating the multi-band formulation And so in this paper we divide things up in a slightly different way compared to the earlier Tone houses are solely papers The what I've been calling the local circulation piece is divide divided into This SR is short-ranged Evo. No, that's not right Self-rotation. Thank you. Thank you self rotation piece So that's basically like the local circulation and then this one is like the itinerant circulation, but The self rotation is defined in a slightly different way. It's basically relative to 1a centers Instead of relative to lattice vector centers and then What we showed in this paper was that this self rotation term number one is gauge invariant And this combination of these other two terms are gauge invariant. So gauge invariant in the sense that Nicola introduced yesterday invariant with respect to a different choice of this unitary mixing matrix u of k throughout the Bruin zone and And then so so this is a way of decomposing things in such a way that you have two gauge invariant terms Which also has some computational Advantages, but it also suggests as Nicola said that there should be some experimentally measurable aspect of a gauge invariant quantity and What Evo found? Was that you could relate this first gauge invariant piece to a certain dichroic sum rule I haven't written it out, but it's basically an integral over frequency of the optical absorption anti-symmetric part Integrated over all frequencies with maybe a power of omega in there I don't quite remember but in any case it's something like an f sum rule, but for the anti-symmetric part of the Conductivity, which is the dichroic part of it Okay, I said I would give a few Experiments, sorry a few calculational results. So this is actually not our work, but Davide Sarasoli together with other collaborators Did a calculation of the orbital magnetization of some real materials iron cobalt and nickel using this theory and This delta M over here is something that I won't talk about It comes from a core correction from the way that we're treating pseudopotentials and so let's ignore that That's very small. What you see is that the local circulation piece is Maybe five or ten times larger than the itinerant circulation piece So the local circulation piece unlike in our, you know, Haldane model. It was like 50-50 Actually, they were opposite signs in that case But but here in in real life the local circulation piece does tend to dominate when you add them together You know, the local circulation would have given you a decent approximation as it happens and then in this paper Shortly afterwards. We did similar calculations, but using Juanier interpolation that Eva will talk about in the next lecture to do the calculation and Well, there's a bunch of results here This is for iron cobalt and nickel the ones that have been grayed out are for magnetization Tilted in some direction that is not the ground state orientation. So let's just focus on the correct ground state orientation Over here are the experimental numbers for the orbital. I'm sorry experimental numbers for the orbital magnetization and This muffin tin thing over here on the right hand side is where you only calculate orbital currents Inside some spheres around the atoms and ignore the interstitial part So that's an easier calculation to do and this is using the quote-unquote modern theory I think Raffaella were we're in an intermediate era where you're allowed to say modern theory if you put it in quotes And then eventually we'll come to a point where we can't say it at all But anyway, so this should be modern theory in quotes And what you see it well and reference 14 is that it's a Sarasota Lee at all And so the two theories are in pretty good agreement with each other. There were different different pseudo potentials different codes But what you can see is that the muffin tin calculation is You know qualitatively correct the right sign the right order of magnitude, but actually the modern theory Numbers are in better agreement with experiment. They're not in perfect agreement either But they're in better agreement on the whole with experiment than the muffin tin numbers. I don't Then there may be more up-to-date numbers for this nowadays, I haven't really tracked the field to know Okay, so maybe what I should do is is stop for a moment and and see if there are questions about this because I'm I'm really kind of going to change gears a little bit to talk about hybrid 1a functions So let's see if I have questions at this point. There's a question over there is the Contribution of each k point and each band to the orbital orientation also physical quantity For example, is it related to the orbital moment of each eigen states? Yes, so the The work of the of the Chen you group addresses that question So if you if you make a if you choose a particular wave vector in the B1 zone And then you construct a wave packet that is centered at that wave vector So it let's say a little Gaussian wave packet So it's localized both in case space and in real space and so you can discuss its orbital moment its orbital moment is So I've divided this into local circulation and itinerant circulation but That local moment is corresponds to a particular linear combination of those two contributions and Then there's another contribution, which is called the density of states effect Which is that if you apply a uniform magnetic field then the the density of states? of Is modified by an amount that's proportional to the Berry curvature, so basically there's a second contribution which comes from the That you can think of as the modification of the local density of states, which usually is just you know in in in D Dimensional space is just 1 over 2 pi to the D That's the density of states in case space But in the presence of an external magnetic field that gets modified So there's a second term which is the local modification of the local density of states, but yes the the first term Which is the the circulation of the wave packet? Does correspond to you know one of the some linear combination and this other other more spooky thing that I just mentioned is a Different linear combination and when you sum them you get the same things so there is in some sense a physical Interpretation of the local object in case base. Let me just see if there's something in the chat No, it doesn't look like it Anything else Okay, we can have questions at the end as well. So let me let me go on then to talk about hybrid 1a functions and so Let me Begin just for a moment by talking a little bit about topology and what I regard as the simplest kind of topology Which is the two-dimensional quantum anomalous Hall? insulator What happens in the case of a quantum anomalous Hall insulator it has to be a Ferro magnet or has to have broken time reversal symmetry in order for this to happen It has the property that if you apply a small electric field in the x direction this curly e is an electric field You get a transverse current in the 90-degree direction this current J and The ratio of the transverse current to the longitudinal electric field is exactly e squared over h Or in general it's a integer times e squared over h where the integer is known as the churn number and So this is like the integer quantum Hall effect, but There's no external magnetic field here. We just have to have a two-dimensional system that Has broken time reversal symmetry that plays the same role that an external magnetic field would play So how does this? kind of topological insulator come about One way of understanding it is by this kind of hybrid space view And what I'm going to do here is do something a little bit unusual We're used to working in real space Let's say this is a real space unit cell with dimensions a and b measured in angstroms And we're used to dealing in reciprocal space. This let's say is a B1 zone Where the lattice vectors have units of inverse angstroms and I'm going to start working in this hybrid space Which is a space in which on the horizontal axis? We have wave vector in units of inverse angstroms and on the vertical axis. We have real space In the y direction, so it's a little bit Difficult to wrap your head around the first time you see it But I think once you'll get used to it. You'll see it's a it's a useful It's a useful trick and so one way to think about what we're doing is let's say we have a two-dimensional Insulator single band insulator again to make things simple And so I've got a mesh of k points in this diagram at the left and what I choose to do is I choose to Look at each individual kx point and think of the string of k points along the y direction as a kind of a one-dimensional system along the y direction and I can one-year rise the system in one dimension I do a one-dimensional one-yearization along the y direction to get local 1a functions and 1a centers in the y direction and what I'm plotting here are the 1a centers in the y direction Plotted versus wave vector kx as k goes across the B1 zone And of course the left and right sides of the B1 zone are really identified so I have to come back to myself and this shaded object up here is really the same set of 1a centers because if I translate I'm sorry. I didn't mean to do that yet if I translate by One lattice vector, which is like Updating the Berry phase by 2 pi that doesn't really change anything So there's another 1a function up here and there's another one up there and so on and so because the left and right sides of the Unit cell are identified with each other You might think this is the only thing that can happen, but you already saw me flash ahead to the next picture That's not the only thing that can happen What can happen is that as you go across the B1 zone the location of the 1a center in the y direction in real space as You go across the B1 zone in kx space may have this kind of winding It may wind by 2 pi or it could even wind by 4 pi or minus 2 pi and That winding number one two or minus one is known as the churn number what I've illustrated here has a churn number of one and You can kind of understand two things from this picture one is that this winding number is a topological Invariant that is to say if I take this Hamiltonian for my two-dimensional system and I Gently modify the Hamiltonian I make some adiabatic change of the Hamiltonian The winding number can't change unless something drastic happens and what that would be is if the system goes through a metallic Phase and then back into an insulating phase Once it's in the metallic phase this picture doesn't apply anymore And when it comes out into a new insulating phase it can change its winding number back to back to something else Sorry, so it can go from here to a metal to here But without going through a metallic phase it can't change the winding number The second thing you can understand from this is in some generic way some hand-waving way why it is that this churn number corresponds to Anomalous Hall conductivity so you know from Boltzmann transport theory that if you apply an electric field Let's say in the x-direction This applies a force on the electrons and the electron wave packets migrate in the kx direction At a not a speed which is proportional to the electric field I'm kind of imagining for the moment that electrons have positive charge So they let's say they go electron wave packets also go to the right as they do that They climb up in real space y-direction So they're actually 1a centers are moving vertically in the y-direction which corresponds to a vertical current and so you get a Sorry, you get a vertical current that is proportional to the horizontal electric field And when you work out the details of that it turns out that it's exactly I didn't write the formula again but it turns out that the Anomalous Hall conductivity sigma yx is just e squared over h times this churn number So in this case, it's exactly e squared over h it basically you just if you work out for the electric field You know one block period is how long it takes the wave packet to go across the Brie-Wen zone in that time It also moves by exactly one lattice constant in the y-direction and you put in the numbers and it turns into just e squared over h Okay, so what have we actually done well here? I'm just focusing on the 1a center positions of these one-dimensional chains of k space of k points, but we can also focus on the Eigen states for which those are the eigenvalues in other words at a given kx I I Construct these 1a functions, but these 1a functions are still extended in the x-direction So that's the picture that I have at the top here. This is a hybrid 1a function h is for hybrid 1a function It has a quantum label kx because it's well-defined at each kx It also extends across to infinity in the x-direction, but it's Exponentially localized in the y-direction So here this is a picture of the block function over here Of course, there's another hybrid 1a function if I choose to 1erize first in the x-direction and then keep ky as a as a wave vector index and then here is the regular the 1a function that's really localized in in both dimensions and Basically to go from the block functions to the 1a functions You just Fourier transform in both x and y-directions, but if you just transform first only in the y-direction And then in the x-direction you go by this top path so Fourier transform in the y-direction gives you these hybrid 1a functions and You know if you look carefully in the upper corner here I wrote the Fourier transform formulas that tells you how these things are related to each other, but I think you get the idea You basically just instead of doing the Fourier transform simultaneously in both x and y you do it only in x or only in y And so the center of charge of this hybrid 1a function in the y-direction at a given kx is what I was plotting in this figure But we can also recover the entire hybrid 1a function itself if we want and that's useful to do for for some purposes So one thing you should appreciate if you are Especially Exposed to the field of topological insulators is that these hybrid 1a centers are one and the same thing as what people call Wilson loop eigenvalues The Wilson loop is something that Is more of a favorite of people from a field theory Formal background and I won't explain exactly what the Wilson loop is but essentially When you do this construction of one-dimensional 1a functions and choose and pull out the 1a centers That's mathematically identical to calculating the Wilson loop eigenvalues Expressed as numbers as phase angles between 0 and 2 pi And the other thing I should say is that this this one-dimensional 1a is very simple I don't think Nicola Dwelled on this yesterday, but it turns out that you don't have to do any projection You don't have to do any iteration You don't have to do any max the maximally localized 1a functions in one direct in one dimension can be computed by essentially straightforward diagonalization techniques and very fast no iteration required and and free of Free of topological obstruction. I haven't talked about topological obstruction yet, but I will in a moment so unless this particular string of k-points happens to hit a Vile point or something like that. You can always construct the hybrid 1a function and our perspective here is going to be not only just to look at the hybrid 1a centers, which is Usually the Wilson loop Perspective, but also the 1a functions hybrid 1a functions themselves are often useful Okay, so let me use these hybrid 1a functions to talk about a topological Obstruction, this is maybe not the only or even the most natural way to talk about topological obstruction, but But I think it's it's one way So here again is what I did before I construct the the hybrid 1a centers for each kx corresponding to the string of points along the ky direction and I plop them like this and You might ask What mathematical formula describes this? Curve which is y bar as a function of kx So all you have to do to answer that question is to take your hybrid 1a functions So this zero means it's the one at our y equals zero and Calculate the expectation value of the y operator as a function of kx And if you plug in the Fourier transform formulas that relate the hybrid 1a function to the true 1a functions oops I'm trying to I'm trying to To do the cursor on the screen instead of with a laser pointer because I know that the people who are seeing this Virtually don't see the laser pointer, but occasionally my finger misses Okay, so where was I okay? So it turns out that the expression is just this you have to take a matrix element of a position operator y position operator between two Ordinary 1a functions one at the origin and one displaced by by x so origin and x you calculate the y matrix element between them Because the 1a functions are exponentially localized This is only non-zero for a few 1a functions that are nearby And so you get a few Fourier components and those Fourier components describe the variation of this Of this curve that I grew up here But there's no way that I could ever possibly get a curve that looks like this from from this kind of Fourier transform right because it's just you know It's just a sum these these matrix elements over here are just some finite numbers the sum only has a few terms in it and Obviously it has to come back to itself And so what this means is that? if These 1a functions exist if the system can be described by localized 1a functions Then the hybrid 1a centers can never have a non-zero winding and Therefore the system must be topologically trivial or conversely if the system is not topologically trivial if it's topological If it has this churn number of one or minus two then 1a functions do not exist it's not possible to make maximally localized 1a functions of this two-dimensional sample and In either case by the way the hybrid 1a functions do exist so the topological Obstruction doesn't obstruct you as far as killing the hybrid 1a functions But it does kill the existence of ordinary 1a functions another way of seeing this is from the point of view of But Nicola mentioned yesterday that normally you insist on a gauge Which is not only smooth and continuous but also periodic and it turns out that that's just not possible To find such a gauge in the case of a topological insulator It's a little bit like the story of describing spinners on the block sphere There's no gauge that you can choose that doesn't have a singularity somewhere The gauge that you see in most textbooks has a singularity at the south pole Okay That's for a certain kind of topological insulator Namely the quantum spin of the quantum hole insulator that occurs in a sample that has broken time reversal symmetry Just briefly I want to mention that when you do have time reversal symmetry You have a different kind of topological insulator in two dimensions, which is called a quantum spin-hole insulator Or a Z2 topological insulator Z2 is just that means either even or odd so it just it's either ordinary or it's topological and The way this you can think about this is that a quantum spin-hole operator is insulator is basically like superposing a quantum anomalous Hall state with a plus one churn number and Quantum anomalous Hall state with minus one churn number this one for the spin-up electrons and the second one for the spin-down electrons and you put them in the same material and That has time reversal symmetry because the time reversal symmetry reverses the sign of the churn number but also it reverses the sign of the spin so the system as a whole comes back to itself and You can then turn on some spin orbit coupling or something and it still has this topological property and In this case, there's no Ordinary anomalous Hall conductivity because the up spin and down spin currents cancel each other But there's a kind of a spin current instead So if you do the same construction of the hybrid Wanier centers as a function of wave vector in a situation like this What you see is something like this you basically have the The up-going curves are the up spins with churn number plus one and the down-going curves are the Quantum anomalous Hall state with down spin with churn number minus one and then they then they intersect if you if you turn on the mixing You do not open these degeneracies at k equals zero and at k equals pi Because those are predicted by Kramer's degeneracy and so whether the system zigzags like this or not Determines whether the system is topologically non-trivial or trivial so this we did for the Kane-Molley tight binding model is a calculation of How these hybrid 1a centers behave as a function of kx in a part of the phase diagram You just tweak the parameters to put the system in a topologically trivial phase or in the topological phase And you see whether it zigzags or doesn't zigzag Is whether the system is normal or topological now? How is this related to topological obstruction? Well, if you try to construct 1a functions for a normal system What will happen is 1a functions will come in pairs there'll be a 1a function on this site and Then a time reversal image of it with spin down Let's say it had spin up the time reversal image of it will have spin down if it had current going clockwise So you make the time reversal image of it and those two 1a functions are orthogonal to each other And then you can construct a whole set of 1a functions that always come in Kramer's pairs if you have Six bands that is to say three pairs of bands, then there'll be three of these pairs of 1a centers in your unit cell And in that case you will always find that you get this normal dispersion of the hybrid 1a bands if you have this kind of Unnormal dispersion of the hybrid 1a bands You cannot have those Kramer's pair So what happens in this case is that the statement about topological obstruction is a little bit more subtle It is possible to make 1a functions for a two-dimensional quantum spin-haul insulator But you cannot do it in such a way that they come in Kramer's pairs You can't do it in such a way that the gauge Respects time reversal symmetry you have to break time reversal symmetry in the gauge and then form maximally localized 1a functions in order to make 1a functions, so it's kind of an odd Situation so for the quantum spin-haul insulator You cannot construct ordinary 1a functions as you would in the way you expect where they come in Kramer's pairs You can construct 1a functions. We showed that explicitly this was with Alexei Soloyanov You can explicitly do it if you're willing to Break the time reversal symmetry in the gauge Okay the last part I want to talk about is what else we can do with hybrid 1a functions and We did some work a few years ago about how these hybrid 1a functions Here I'll be focusing mainly on the just the hybrid 1a centers are a useful tool for visualizing different kinds of topology So Here what we're doing is we're 1a rising We start in three-dimensional case space and we pick one direction the z direction and we want to 1a rise in the z direction So there's another kind of hybrid a double hybrid, right? You could go and 1a rise in two out of the three dimensions, but that's not what we're doing So we want to arise in the z direction We still have kx and ky in the horizontal directions and so for each vertical string of points we calculate the the 1a centers along that direction and if you plot those 1a functions as a function of kx and ky You get pictures that would look something like this. This is the Brewan zone on the XY plane on the bottom and and then again the vertical direction is real space z direction and here are the These 1a sheets and they have to be of course periodic in kx and ky and this would be for a two-band model because there are two 1a functions per unit cell in the y direction and this would be for an ordinary trivial There's nothing special going on here Since these two sheets don't fall on top of each other this this would be for some magnetic system if it were Non-magnetic then probably the sheets would be simpler So here's what happens if you have actually here's what happens if you have time reversal symmetry So this would be a non-magnetic two-dimensional insulator. It turns out that the 1a sheets are forced to touch at Four points which are known as the time reversal invariant momenta So it's zero and pi in the kx direction and zero and pi in the ky direction And that comes about from a kind of a Kramer's degeneracy applied to the hybrid 1a sheets and And so what you can do is if you think about the So what happens is that? In the time reversal invariant system It's useful to focus on just one quarter of the Brion zone instead of the full Brion zone So here's a picture of the full Brion zone k space in all directions, but this is like one eighth So you really focus on one quarter In the kx and what so now the horizontal axis only goes from zero to pi the Ky only goes from zero to pi. This is the same picture as I had here But I'm just showing one quarter of it because by looking at just one quarter. I can see what's going on so this is an ordinary time reversal invariant system where At an arbitrary k point in the two-dimensional Brion zone the wave functions at that k point Don't have time reversal symmetry. They're related to some wave functions at minus k And so these two sheets split from each other But they meet at the corners Now what's a topological insulator? Here's an example of what's called a weak topological insulator where this would be for a system that has two occupied bands The Kramer's degeneracy requires that these bands touch at the four corners But instead of touching in a way that does not involve any Any zigzagging here they zigzag in the y direction if they zigzag in the y direction But not in the x direction or vice versa. That's called a weak topological insulator You can look at you know If you look at a given sheet and label whether the sheet touches from the top or from the bottom to the Kind of a direct point in one a band space Then there are two tops and two bottoms in this case and then a strong ti what happens is you have something like Three three tops and one bottom or three bottoms in one top That turns out to characterize a strong topological insulator. So these hybrid 1a sheets provide a way of thinking about Topological systems by the way, I didn't show this but if there's a vial point then these sheets sort of spiral around the vial point In the sense of a Raimond Raimond sheets So this was work that was done mainly by my student at the time Mariam to Harry Nahad and What she did she did some tight binding models, but she also did calculations for real materials This antimony selenide is a normal member of this class of materials that includes the well-known strong topological insulators Bismuth selenide and bismuth telluride and she actually calculated these 1a bands and You don't when you do this for antimony selenide. You don't see much You just see these basically are located the 1a centers of p orbitals and so on This is going around the four sides of that one quarter B1 zone and Here are the plots of what these 1a sheets look like not much going on when you do this for bismuth selenide instead Look over here. You see that these you get these zigzagging So actually these 1a bands do zigzag in In one corner of the of the B1 zone if you look at the band structure Energy versus wave vector you can't tell any difference by looking at the band structure of between bismuth selenide and the antimony But when you look at the 1a sheets, you find that they have these touchings at the B1 zone corner so and then There were some other examples that we did of other kinds of like mirror churn insulators and so on where you can see something about The topology by looking at the hybrid 1a centers As a computational tool so the last thing I'll do there's just two slides and I'm going to not go into Into the mathematical detail, but I just want to give you a feeling for this So again with Mariam and then also with Thomas Olson together with Evo We had some papers where we looked at what happens with these hybrid 1a bands. That is the plots of 1a centers versus the two-dimensional Bruin zone and So here what we've done is we've actually defined a very connection on the hybrid 1a bands So in other words a very connection of the hybrid 1a functions So each hybrid 1a function has a label L Which is the unit cell in the vertical direction and n which is which of the 1a functions within the cell in the vertical direction And then it actually has an argument Kxky which is Hidden but of course the very connection involves one derivative and it's written this way and the Berry curvature involves two two derivatives and it can be written this way And so each one of these 1a sheets has a churn number that you can not define by integrating over the over the Berry sheet and So for example the total anomalous Hall conductivity can be written as some of these hybrid 1a churn numbers and There's something that I don't have time to explain But if you have run into it before there's something called the churned Simon's axion coupling That gives you a kind of a formal Contribution to the isotropic magneto electric coupling and we also found that it was possible to write the this churned Simon's Axion coupling in a rather a pretty way in terms of the 1a sheets Basically you integrate over each 1a sheet The Berry curvature times its Z-coordinate position And then there's another correction here, which has to be included but the topological piece is really this first piece and so to do this you need not just the hybrid 1a centers, but the actual hybrid 1a functions and their very connections and Berry curvatures and Another project that came out of this was looking at Something called surface polarization. So imagine that you have an insulating Three-dimensional material that has inversion symmetry in the bulk But the surface Has low enough symmetry so that the surface can have a non-zero polarization parallel to the surface How do you calculate that? polarization parallel to the surface and what we did in these papers was basically construct the hybrid 1a sheets in a slab calculation and calculate the Electric polarization, which is the Berry connection integrate the Berry connection in the in the kx direction to get the polarization in the x-direction Calculate the contribution of each one of these 1a sheets to the polarization and you find that there's some excess x polarization at the top surface and maybe Depending upon the symmetry, maybe also at the bottom surface or maybe the opposite sign at the bottom surface so So We have come across a couple of situations where these hybrid 1a functions are actually a useful tool for Deriving quantities that otherwise it would be hard to know how to calculate So that takes me to the end. I have time for I guess a few more questions The top the part about hybrid 1a functions I use the hybrid 1a functions as a way of talking about topology and topological obstruction But also as a way of visualizing topological systems and a little bit as a computational tool for Looking at for example surface topological properties Okay, so I'll stop there and see if there are more questions Thanks for the nice talk. I'm pretty new to this field. So maybe this knife question, but you were speaking that for construct in this hybrid 1a function you need localized like Exponentially localized manner function. So for that you need a gauge field which Like allows you to do that and you also spoke about like you have to break time reverse out to do this in the case of set to invariant or whatever, but Does this always happen like can you always define a smooth gauge for this? like compiling with whatever or Would you need to break further like symmetries or whatever? No, so the the thing is that when you In the hybrid 1a construction, we're always doing the one-dimensional one-yearization and so Mathematically, it's the same as if you had a polymer So a one-dimensional system with some periodic repeat and you're constructing one-dimensional 1a functions and So in this case If you remember Nicola introduced the overlap matrix So you take the you take the cell periodic block functions at this k and take the inner product with the set of them at the next K so if you have if you have let's say four 1a functions per cell or four bands Then you're gonna have a four by four matrix and then another four by four matrix And then another four by four matrix and then there's just some four by four matrix algebra what you do is you multiply all these matrices together and Then you do some singular value decomposition on the product and you extract the eigenvalues and there it is So it's a completely straightforward operation in the one-dimensional case the problem with maximal localization is because You have to try to maximally localize Simultaneously in x and y directions or x y and z directions, and then you have some competition if you all you want to do is is Maximally localized in the z direction. It's trivial. There's no competition. So it turns out that it's really very straightforward You don't have to do any real tricks so when you say that you need to break time reversal symmetry in order to get Exponentially localized by your functions in for example in the example of the c2 odd the topical insulator What's the actual meaning of breaking time reversal symmetry on on the basis because I mean the system still preserves time reversal symmetry So when I say that the system has broken time reversal symmetry I mean that it's a magnet. It's a ferromagnet or some some in some cases anti ferromagnetic or ferrimagnetic states will do it So what I mean is that time reversal is spontaneously broken right in the same way that if you have a crystal That's a para electric and then it goes through a phase transition and becomes ferroelectric Well, this universe still has inversion symmetry, but the system has broken spontaneously broken. It's It's inversion symmetry. So I'm talking about magnetic systems in which time reversal symmetry has been spontaneously broken so You can only have The quantum anomalous whole state when you have broken time reversal symmetry Which you can see pretty easily because even for a metal if you don't have time reversal symmetry You can't have a transverse current because if you imagine reversing the reversing, you know playing the movie backwards in time it wouldn't work and so So you have to discuss the case where the where the material has time reversal symmetry in the case where The material doesn't have time reversal symmetry as two different Situations and study the topology and in those two situations Independently in the case where time reversal symmetry is broken. You can have this Churn insulator state in the case where time reversal symmetry is present. You cannot But there's a new topological index the z2 index that you can define that is not defined in the magnetic case So it's just two different cases that you have to consider separately Sorry, so I probably I didn't explain myself. Well, so what I meant is that in the c2 all the case Yes, when you try to construct the maximum illocalized linear functions, okay Because you said that you need to break time reversal symmetry on the gates Transformation in order for the the binary functions to be maximally localized. Yeah So, what's the the meaning of breaking time reversal symmetry because I mean the system still preserves it, right? So again the way to think about it I think best way to think about it is in terms of those you matrices that Nicola was talking about so you can go let's say From from from from one gauge to another gauge by introducing this Let's say I have four bought four bands So it's a four by four unitary matrix is a function of k and if the matrix at k is the Hermitian conjugate of the one at minus k And you enforce that everywhere in the B1 zone as you make your gauge change That's a gauge change that preserves time reversal symmetry so for example if you Calculate all the states in the half B1 zone and then in the other half of the B1 zone You define them to be the states at k the state at minus k is the one at k times the Hermitian you know conjugate That you start with a time reversal preserving gauge And then you apply you matrices that only have this property that the ones at k and minus k are Hermitian conjugates of each other Which for unit areas inversions of each other then you then you preserve a Then you preserve a time reversal invariant gauge So that's that's what I mean and that when you think about it what that corresponds to is what I said Is that the 1a function come come in Kramer's pairs? That that's basically the meaning in real space of what I just said in k space There was a question on the chat if 1a functions exist the system has trivial topology Is this statement only true for 2d systems or any dimensional system? well again, you have to talk about different Symmetry classes separately, but if we talk about the case of broken time reversal symmetry where you can have a churn number in two dimensions You have a single churn number corresponding to the Berry curvature in the xy plane in a three-dimensional Insulator in principle you can have three different churn numbers corresponding to x y x z and y z if any one of those three churn numbers is non-zero then you cannot construct exponentially localized 1a functions for the insulator But then you can now go talk about the time reversal in variant case or other cases Let's maybe not go there Are there in general problems to define 1a functions for other kinds of topological states? Yeah, so the basic paradigm goes through suppose you have a mirror churn number a mirror churn number. I'm sorry This is a kind of topological crystalline insulator where Let's say on a particular Kz equals zero plane The states with odd mirror symmetry and even mirror symmetry have equal and opposite churn numbers That that can happen and that's called a mirror churn number And then you can show that you cannot construct 1a functions that have mirror symmetry in the way that you would expect So the general general statement is that if there's some symmetry that protects the topology And you're in a non-privile State of that topology, then you cannot construct 1a functions that obey the symmetry in the way that you would naively expect Okay. Yeah, we're getting a little over time So, you know You can get for polarization, for example, you can have these Scarmion like structures in certain systems and that would have a you know non-zero topological Charge, so do you think for those systems? You also have these problems of not being able to have the vanier functions, you know Which I guess you could describe the local polarization from I think that's not a short answer But So I think it's very so what you're talking about is topology that happens in real space And what I'm mostly talking about is topology that happens in case space And so you could talk about systems Let's say you put your skirmjons on a skirmjons lattice And now you have topology that happens in both real space and in case space and how does that play out? Good question. I don't have a short answer. I want to follow up about vile sentimentals. You said that the Vanier sheets start to curl and kind of Go along The spiral so does it mean that you also have problem with vanierization of something like thallium arsenide? Well, but gallium arsenide. It's not a vile set. What did you say gallium arsenide thallium? Thallium arsenide the vile semi metal Yeah, so so again if you have a vile semi metal and you want to want to arise Only the valence band states by which you mean when you come to a vile point only the ones below and not the ones from above then you have a Singularity in the description of the block functions at the vile point and that singularity means that when you try to construct Wanier functions, they cannot be exponentially localized. They have power law tails So you can't make one a functions for a for a vile for a vile system But if you include more states than then you could or well I mean, yeah I mean you can you can include some of the conduction bands and then what you do is you want to arise The larger space and you use the one a functions as a basis for describing The band structures and then you'll see so that's what one a interpolation is about which you'll hear about in the next talk So that's a good segue. I guess to you Eva. Okay. Thank you So Eva is going to use my laptop. So let me see if I can get this set up So let's thank again, baby