 Good morning. I'm Shobna Narasimhan. I'm from the Jawaharlal Nehru Center for Advanced Scientific Research in Bangalore, and I'm chairing this session, especially given the topic of this session, which is topological materials and phenomena. We thought we'd pause a minute to mourn the unfortunate passing and also celebrate the life of this person who is Shu Cheng Zhang, who contributed so much to the themes of this session and was a brilliant physicist. I did not know him personally, but I certainly admired his work. He had many links also with ICTP and this photograph was actually taken when he won the Dirac Medal. He was one of the co-winners of the Dirac Medal in 2012. The Dirac Medal is a very prestigious award given by ICTP. He also organized conferences at ICTP. For example, this one, which is it's on high-dimensional quantum hall effect, Chern-Simon's theory in non-commutative geometry and condensed matter physics and field theory. The interesting thing to note about this conference is it was in 2005, so it was a year before topological insulators exploded onto the field. So the Kane and Melly paper and the Bernoulli and Zhang papers were in 2006. So this conference was like this one organized at ICTP to introduce the themes of the two talks today. I thought I would use two quotations from Xu Cheng Zhang. He said as a student, I was deeply attracted to Dirac's style of searching for physical laws of nature by looking for beautiful mathematical structures. And he also said, once you have a topological way of describing physical phenomena, there's a huge mathematical literature that you can exploit. And in his Dirac Medal lecture, he mentioned that he started out life as a particle physicist working on field theory and then he started getting very frustrated because he thought that the theories that he was coming up with, he couldn't find experimental tests for them. And then he went and talked to Xi'an Yang who suggested to him that maybe he should turn his attention to condensed matter because in condensed matter, it's much easier to find experimental realizations. And so the two talks today show both aspects of this. They show that if you use these theories that were initially developed either in mathematical physics or in high energy physics and field theory, you can discover very beautiful properties of condensed matter. And also somewhat surprisingly, condensed matter physics can serve as a test bed for theories of phenomena that you normally think of as happening at high energies. So without further ado, I think I'm not really an expert in these topics. So I think it is better if I turn it over to the two speakers. So we have two talks in this session. The first talk is by David Vanderbilt who I think doesn't need any further introduction to this audience. He's been coming to this conference for many, many years, I know. He's at Rutgers University and the title of his talk is Axion Coupling and Magneto-Electric and Topological Materials. Okay, I think we're in business. Thanks, Shobana. It's wonderful to be here. A lot of old friends. It's an honor to open the conference. What I want to do in this talk, there's a huge development in topological materials, topological insulators and semi-metals in the last decade and I can only take a little bite but the piece I want to introduce is the so-called axion coupling and what gives rise to so-called a kind of topological insulators known as axion insulators of which currently we don't know any experimental examples so you could regard that as a disadvantage or as a challenge. I obviously prefer the second point of view. Before I forget to advertise the many collaborators who have contributed, who are co-authors on various points of papers that I will mention during the talk, here are a few I won't read the names but many of you probably recognize many of them and I'll mention the papers as I go along. So what I want to do is do a very brief introduction and review of baryphasis and electric polarization which I hope most of you have some degree of familiarity with and then I'll talk about anomalous Hall conductivity and then the axion coupling theta which is closely related to the baryphase phi but in three dimensions instead of one dimension and the physics that it's related to which is a magnetoelectric response instead of electric polarization and then I'll talk about axion insulators and assuming I have time I'll talk a little bit about some recent work that we've been doing with my student Niko Varnava about calculations on tight binding models of axion insulators. As I mentioned, we don't have any good physical realizations in the lab yet so we're working on models. So I hope this gives you a sense of one piece of what's going on among many in the topological literature. Okay, so a baryphase is defined for a one dimensional insulator. I've drawn the one dimensional bryuan zone here as a loop instead of a segment because it's pi and minus pi are identified and I've plotted the energy band vertically and so the energy band is a loop and a loop supports topology or at least a geometric phase. You can define a geometric phase in this way. Unk is the cell periodic block function and from this derivative with respect to k you define the baryconnection and then the baryphase is basically an integral of the baryconnection over the bryuan zone. There's a gauge dependence which means if you put in a phase twist of the Unk and k does change but the baryphase does not change except some integer times two pi so it's well defined modulo two pi, it's a phase. And what is it useful for? Well, let's go up to two dimensions and suppose I look at a two dimensional bryuan zone in terms of kx and ky and for each kx along the ky direction I calculate the baryphase and plot it and here I plot two images because they differ by two pi. Either branch would be equally sensible and if I take the average value of the baryphase across the entire bryuan zone what that tells me is the component of the electric polarization in the other direction in the y direction. In fact, most of you probably know that these baryphases are related to the positions of the YNA centers so you can regard this as a framework in which we keep kx as a block wave vector labeled in the x direction and then we wanyarize in the y direction so called hybrid wanyar representation. So you can also think of these as plots of the wanyar center position in y as a function of kx and if you average these then you get the polarization in the vertical direction for a three dimensional crystal of course you have to average in kx and ky so here phi is really the average of this baryphase over kx and ky and there's a surface theorem that tells you that the surface charge that you find at the surface providing the surface is insulating must be given by exactly this polarization plus an integer and the integer can be regarded either as in the two pi ambiguity of phi or more physically you could add a layer of charge one electron per unit surface unit cell that would change the surface charge by that same amount. Okay there's another strange thing that can happen in rare circumstances you have to have broken time reversal symmetry so this has to be a two dimensional ferromagnet and in that case since the left hand side and the right hand side of this one dimensional B1 zone are identified with each other whatever value the baryphase has here it also has to have here but it doesn't have to come back to itself without wrapping it can wrap by two pi or by minus four pi or by some integer times two pi and if that integer is non-zero then this is a topological system it's topological in the sense that you can't adiabatically connect something that does have a wrapping with something that doesn't have a wrapping without a bulk gap closure in the two dimensional material and so this integer is called the churn number and what it's physically related to is the anomalous Hall conductivity of this material so most materials that don't have this wrapping if it's a two dimensional insulator would have to have a zero anomalous Hall conductivity but there are exceptions where it can be an integer and if it's an integer the reason why it's connected with the anomalous Hall conductivity here I just give a quick physical idea if you imagine applying electric field in the X direction so what that does is it applies forces to the wave packets and the wave vector KX migrates to the right and as that happens the wave packets their Y coordinate the very the 1A center in the Y coordinate migrates in the plus Y direction which is a current in the Y direction so if I apply an electric field in the X direction I get a current in the Y direction even though this is a gap system so in that sense it's an insulator but it can support this kind of anomalous Hall current since 2013 we've had experimental realizations of this two dimensional quantum anomalous Hall state starting basically all of them that we have today are low temperature realizations in magnetically doped topological insulator systems so they do exist we'd like to find more robust and higher temperature versions of them and that's one of the things that we've been working on off and on in my group but for the purposes of this talk I want to go on and talk about not the two dimensional anomalous Hall conductivity of a film or isolated two dimensional system but the surface of an insulating three dimensional crystal so I can ask the question so what I told you before is that for an isolated two dimensional system the anomalous Hall conductivity has to be zero or two pi times an integer and so you might think the same would be true at the insulating surface of an insulating crystal so in other words you apply an electric field and you measure the transverse current at the surface and you might expect that has to be quantized like that and it turns out that's not true this is an example of what is called an anomaly something that can happen at a surface of a higher dimensional system and cannot happen in an isolated system and the way that you usually get out of an anomaly is that there's something else somewhere in the system that compensates and so what happens in this case is if you apply an electric field you get equal and opposite currents on the top and bottom surface if you make a film out of this the total anomalous Hall conductivity is still zero so it doesn't violate the two dimensional theorem that I told you before but you have currents at the surface and in fact what happens is you get currents all around the surface and in fact what this is is an orbital magnetization and of course you can have an orbital magnetization in an insulating three dimensional crystal and it drives surface currents that are given by you know by m cross n hat is the surface current k and so basically the orbital magneto electric coupling and the surface anomalous Hall conductivity are the same thing from this point of view. Okay, the magneto electric coupling I just wanna briefly mention is more complicated than I'm going to be treating in this talk in general it can be defined either as the derivative of polarization with respect to magnetic field or magnetization with respect to electric field you have to have a broken inversion and broken time reversal for this to be non-zero in general it has lattice contributions coming from atomic displacements which we will ignore in the frozen ion limit it has both spin and orbital contributions will ignore the spin contributions and just focus on the orbital frozen ion magneto electric coupling even then it has several components here's the total magneto electric tensor up here there's this green part which we called in our papers either Kubo terms or cross gap terms these are more or less ordinary terms that can be written it's not obvious the way it's written here but as energy you know perturbation expressions with energy denominators and but those are non-topological and mainly I'm going in the direction of topological terms so in practice for real materials where all of these effects are small this may be larger than the red one but I'm going to be interested in cases where this red one can become very large so the other term is a purely isotropic term we write it as theta over 2 pi times e squared over hc and then this is the isotropic tensor and this theta, this geometric factor is written here let me write it here where I again remind you of the definition of the berry connection actually this is a berry connection matrix if there are four occupied bands this is a four by four matrix there's a trace of this product of this is called the Churn-Simons 3 form actually Xu Cheng Zhang was the one who first introduced this formula in the condensed matter context although it was well known in certain contexts in the high energy field okay so here's the thing this integral, the integrand is called the Churn-Simons 3 form the integral gives you what's called the axion coupling theta again the integrand is not gauge invariant if you do a phase twist or a unitary twist among the occupied states however the integral is gauge invariant modulo 2 pi in exactly the same way that the berry phase is only well defined modulo 2 pi now in the case of the berry phase it's normal for the berry phase to be on the order of pi or 2 pi that gives you the you can have large polarization in some materials in these magnetic materials these magnetoelectric materials that we're talking about this axion theta is usually very small so for example we calculated what it was in chromium oxide and it's a couple of orders of magnitude smaller than the other spin and lattice contributions in chromium oxide and smaller than other these are other materials that are known to have large magnetoelectric couplings but if this theta were on the order of pi then it would give rise to a magnetoelectric coupling which is among the strongest magnetoelectric couplings of materials that we know so if we can get up to theta equals pi by some topological magic we jump into a region that's very interesting and so that's where we're that's really where we're going okay so I mentioned that the surface anomalous wall conductivity is really the same thing as the magnetoelectric coupling there's a minus sign because well in my convention when you apply an electric field in this direction the orbital magnetization is defined by the right hand rule of the current flowing around but the surface anomalous hall conductivity would be positive if the current were going in the other direction but it's basically the same thing these two things when you've thrown away those Kubo terms all of these things are identical and from a physical point of view we can understand why it is that this theta coupling makes sense that it's well-defined modulo two pi because if I start with some material this white block of material inside it has some initial coupling theta and then I glue onto the surface some quantum anomalous hall layers each of which carries E squared over H of anomalous hall conductivity then you've effectively increased the total magnetoelectric coupling of this block of material by taking theta to theta plus two pi so that's a physical explanation about why it makes sense that this mathematical fact that the theta coupling is only well-defined modulo two pi okay so here's the analogy between baryphase and electric polarization and the theta cup which is basically a one-dimensional problem although here I draw it in three dimensions but kx and ky are basically spectators and this is a fundamentally three-dimensional phase object in this case sigma is the surface charge and for insulating surface it's given by the baryphase plus an integer and in this case the surface anomalous hall conductivity for an insulating surface is given by the bulk contribution theta plus a possible integer so there's a very strong analogy mathematically speaking you skip dimensions so mathematically speaking there's another phase angle in five dimensions and another phase angle in seven dimensions but I don't know what they are like and I guess they're not useful to us okay now what about topology and symmetry so what happens is that if you have either inversion symmetry or time reversal symmetry in your crystal then you'd expect theta to be equal to zero because it's basically a magnetoelectric like property and either one of those should vanish normally if you have time reversal or inversion however because it's well-defined modulo two pi there's a way out theta can either be zero in which case the symmetry is mapped theta into minus theta of course zero goes to zero that's fine or theta can be pi then pi gets mapped to minus pi but minus pi and pi are really the same thing because it's only well-defined modulo two pi and there are two well there are more than two but you can combine these things with other rotations and so on but in the simplest case if you have time reversal symmetry that quantizes theta to be zero or pi and so does inversion symmetry so in the case of time reversal symmetry in a material like bismuth selenide which is one of the well-known strong topological insulators it happens that yes time reversal yes theta has the value of pi and so that means that in some sense and I'll explain that in a moment you would expect a half integer surface anomalous Hall conductivity also in the case of inversion so here I have in mind a system that has inversion but not time reversal so a magnetic three dimensional magnetic material with inversion symmetry but not time reversal symmetry you also have the possibility of theta equals pi and so we know that there's an existence proof for a material like bismuth selenide that this is possible and so we've jumped up to a very strong magneto electric coupling in some formal sense but can we make it behave that way in the real laboratory so again I'm reminding you that theta equals pi means that there should be half integer surface anomalous Hall conductivity actually each surface could have its own anomalous Hall conductivity like minus a half plus a half plus three halves it just has to be one half plus an integer multiple of the quantum and if we can create materials that behave like this it'll be quite interesting I'll show you some reasons why a little bit later okay so I said this can happen either with time reversal symmetry or inversion symmetry the case we know about which is strong topological insulators is time reversal symmetry and there's a paradox here at first sight you would say well the surface anomalous Hall conductivity should be non-zero because it's given by this bulk theta which is non-zero on the other hand if I make a crystal light like this and it has time reversal symmetry including at the surfaces then I know that there can't be any anomalous Hall conductivity because that would break time reversal symmetry so what's going on and what's going on here is that what I told you before was for insulating surfaces if you have metallic surfaces there's another term that you have to add both to the surface charge so for the surface charge you have to add a term which is an integral over the two-dimensional bulk real-en zone if there's a Fermi pocket you have to add up the total charge contributed by that occupied Fermi pocket F is just the occupation function and for the anomalous Hall conductivity you again look at that Fermi pocket and you integrate up not just the area of it but you multiply times the Berry curvature so omega here is the Berry curvature and that gives you a contribution to the surface anomalous Hall conductivity so this is quantized part plus a non-quantized part coming from the metallic surface now what happens the way you get out of the paradox in a time reversal preserved system so strong topological insulator is that these effects cancel each other so from the bulk you get this formal theta equals pi contribution to the surface anomalous Hall conductivity or equivalently to the magneto electric coupling but from the surface you also get pi and the reason is that on these topological insulator surfaces time reversal surfaces there's a Dirac cone there has to be for topological reasons a Dirac cone when you integrate up the Berry curvature inside that little Dirac cone it's equivalent to calculating the Berry phase around the Fermi loop and that's pi it's exactly pi because of time reversal symmetry so this is pi and that's pi and the two things add up to be two pi which is equivalent to zero and so the total surface anomalous Hall conductivity is zero and that's good but what it means is if you have a material like business solenoid it does not act like this magneto electric behavior at least not unless you break time reversal symmetry at the surface for example by interfacing it with a magnetic material so there've been some attempts to do things like that but you have to engineer a new behavior in order to make this kind of material exhibit a large magneto electric coupling on the other hand I wanna talk about so-called axion insulators are the ones in which the theta equals pi is protected by inversion again the bulk theta gives a contribution of pi but the surfaces can naturally be gapped the reason the surfaces can be gapped is that the symmetry that's protecting the bulk topology is inversion but inversion is never a good symmetry at a surface because of course there's vacuum on one side and not on the other and so it turns out that it's very natural for axion insulators if we can find them it's very natural for them to have insulating surfaces and therefore for the surface anomalous hole conductivity to reveal itself as plus and minus e squared over h with no extra work with no extra engineering and that's why this is an interesting avenue to pursue so if we can find these axion insulators they have theta equals pi the surfaces will naturally be gapped each surface will carry a half integer quantum anomalous hole response the big problem is that we don't have any examples of these yet there's one example which is almost like this which is manganese bismuth II at tolerium IV which is an anti-ferromagnetic topological insulator I'll say a bit about that but the big thing is we need new materials five minutes, good. Okay, here's about the anti-ferromagnetic topological insulator this came from a paper by Mong, Esen and Moore some years ago where they imagined stacking alternate quantum anomalous hole layers layers having alternate turn numbers in a kind of anti-ferromagnetic material and they showed that theta equals pi was possible in that case and in the last few months there's been a spew of papers four or five papers about this manganese bismuth II at tolerium IV which is a kind of example it's a little bit like the bismuth selenide materials but there's an extra layer of manganese all these are tellurides but manganese telluride in the middle and then one layer has spin down the next layer has spin up the next layer has spin down and actually what protects theta in this case is time reversal composed with a half translation along the vertical direction that's also enough to reverse the sign of theta and what they find by doing surface angle resolved photorelectron spectroscopy is that there is a gap common to both the gap common to both the bulk and the surface and if they do some kind of high resolution processing to get this image cleaned up I think it's a second derivative of some kind what you see the big black cones are the bulk bands the bulk valence band and the bulk conduction band and then inside there's a direct cone which would have been crossing all the way through the middle if the material were not magnetic but because of the magnetic property there's an opening here just a little gap opening here and if you could get the Fermi level right in there then the surface should have an almost all conductivity of half the quantum unfortunately the Fermi level is up here so this is again one of these systems where there's a lot of defects and the chemistry hasn't been improved to the point where we can bring the Fermi level down where we want this material by the way would have insulating surfaces on the top but not necessarily on the sides because the time reversal times half translation would naturally be a good symmetry on the side and that would preserve the vanishing of the anomalous all conductivity on the side surfaces which requires them to be magnetic so in the last couple of minutes let me just talk about how you might observe this behavior and what you might do with it there's edge channels and the magnetoelectric effect there's also a direct way of measuring the surface anomalous all conductivity optically which I won't talk about anymore but I think that's also an extremely promising direction so in order to get a better idea of exactly how the various surfaces of an axion insulator what determines for example the sign on different facets what we did with Nico was to take a simple model of a pyrochloriridate I won't go through the details since I'm obviously a little bit behind time but in this model the real pyrochlorys lie up here as trivial insulators but there's a region this blue region where we get an axion insulator phase so we decided to play with this model here the blue curves are the surface states on the top the on the bottom the red ones are on the top so you can see that on a given surface you have a nice gap a reasonably nice gap at the surface we used a method that was developed by Ivo Souza and coworkers for calculating not only the total surface anomalous all conductivity but it's layer by layer resolution so we can see how it approaches one half as you include more and more layers near the surface and I won't tell you everything we find I'll actually just tell you this is one of these things where you do the calculation and then afterwards you realize you should have understood this from the beginning what happens is that if you have a crystallite and two faces of the crystallite like the top one and the bottom one are related to each other by inversion symmetry then they have anomalous all conductivity in the same absolute sense that's the meaning of these arrows however we're going to adopt the convention of measuring the anomalous all conductivity from the inside to the outside and in this case these have opposite anomalous all conductivities and I'll color them in opposite ways and then the total magnetic point symmetry of this Pyrochlor is such that if you go to any neighboring facet the surface anomalous all conductivity reverses and then here on one facet to the next facet you have a change of the surface anomalous all conductivity by one quantum it means you have to have a chiral edge channel running along every single edge like this chiral edge channel is like what you find at the edge of a quantum anomalous all insulator it's a one-way channel only if you, so this was for the anti-ferromagnetic state if you study the ferromagnetic state you can play some cute games for example if you tickle the magnetic state in such a way that the magnetization is in the x direction then your chiral edge channels run this way and they connect the vertical wires if you turn the magnetic field then they go this way and they connect the horizontal wires so by flipping the magnetic field you have a little quantum switch where you can a double pole, double throw switch there are a lot of other games you can play if the system back is in the anti-ferromagnetic phase if there's an anti-ferromagnetic domain wall then where the domain wall hits the surface then you have a chiral edge channel if you have a single height step on the 001 surface it turns out that you have a chiral edge channel and these can be used to carry dissipationless currents and so on so if you can really do this it might be very interesting so one application is to engineer these chiral edge channels into your material and use them the other application is to get rid of them so if you want a material that behaves like a strong magnetoelectric you don't want to have different facets that have different colors different happen to your anomalous hole conductivity because then what happens is you apply the electric field and the surfaces currents are going in opposite directions by the way it looks like you're violating charge conservation here but that's not the case actually what happens is because there's a chiral edge channel here charge runs in and takes care of the charge conservation but at least on the 001 surfaces of the pyrochlor system what you can do is add a half layer I mentioned before that adding a half layer changes the color so if you can come into this system and spray on an extra half layer this is the kind of thing that theorists love to tell experimentalists to do spray on an extra half layer of material you basically change the sign of the magnetization at the surface and now you have a consistent flow of surface current all the way around and so now this thing has this very large magnetoelectric coupling it generates a magnetic dipole field which is as strong as some of the strongest magnetoelectric materials that we know about okay so that I think I probably just squeezed into the water under the wire that's my general introduction to this field there's a lot that I left out but I tried to emphasize the formal similarity between the Berry phase theory and the axion theta talked a little about materials but there's a huge need for new materials and if I can just have 10 seconds left I want to do a little shameless advertising here I've just had a book published on Berry phases and electronic structure theory that talks about polarization magnetization 1A functions, axion insulators and many other things and over here on the table I have some flyers that have a 20% discount code on them so sorry about taking a little commercial but otherwise thank you for your attention and I'd be glad to answer your questions