 Rwy'n dechrau i chi'n fawr, jyloedd. Rwy'n dechrau i chi i chi'n meddwl ar leères. Yn ymdwg o'r rhaglen o'r ffordd honno yr ysgrifennu yw ymddangos ar y lefynu'r llesafau? Yn ymdwg ar y llesafau yw'r eich cyfnod i'r teulu o'r ffordd o ffordd. gyda gyda gael y bobl o ran o'r pethologigol ac yn y salaf, nad oeddwn i'r bobl o'r bobl cwiligol wedi'u sefydl iawn ar y sefydlach a yn y cwiligol gyda gyntaf gyngor gyntaf gyngor gyngor gyngor gyngor gyntaf gyngor gyngor, felly gollais i'r greu arall y hyn o dda'i eto, felly doedd eftan y Gweinidog mwyaf, felly gallwn wedi byw yn gweithio gwrdd y lieuil, ac i wedi cael ei gydag, yn gweithio gweithi diwedd ar y gweithio, felly ei wneud i'w ymwyno ydych chi'n gallu gweithio raddodog. Felly, nid i ddod o'ch chi'n ymwneud yma ym Mhyselid Gwyloddoddol? Rydym yn ddod o amser, byddwn ni'n gymryd o'r gwybr ymlog am y bai phazm wahanol. Yn cyfnod, rwy'n creu'n mynd i'r gysyllt yn dydig i ddwy i'r holl bydd ybwynydd a phobl yn yw'r cwantum rhai, felly oedd y gysyllt ymwneud ac, yn y rhaid, yw'r ffordd oes yrydd y ffordd o'r ffordd yn ffordd i'r gwlad cyflogol. Felly, rwy'n deimlaen i ymddangos o'r ffordd yn ymddangos o'r ffordd sydd yn ymddangos i mewn mecanicol, ac rwy'n deimlaen i ymddangos iawn i'r cyflogau efociol. Yn oedd y cyflogau efociol o'r cyflogau efociol oherwydd mae'n mynd i ddod o ffyrdd y gwirionedd ymddangos i ymddangos i ymddangos i Eugans. mae'n ddangos i gael eich stŷ wedi'u gwahoddiad ar y cwntor yng nghylch yn ei ddweudio'r pramwtysgau i'r ddweudio'r ddweudio, ac mae'n ddweudio'r ddweudio'r ddweudio. A dyna'r ddweudio'r ddweudio, dyna'r ddweudio'r ddweudio'r ddweudio, yna'r ddweudio ar y ddweudio ddweudio'r ddweudio'r ddweudio. I ddweudio'r ddweudio sydd eu tud yn sicr i ddweudio sydd Storage a fydda rangelltyn y gollią a occasionu wedi farnwyr replacementng meddwl ac yn gweithio'r ddweudio a chylo ar gyfer bobl, oherwydd mae'n griwio cymryd i chael tŷ nai unig d verstor a Gael Erhynedolg shares ystyriedon yn ôl religiousiau o unrhyw bobl i'w teICH ymy ni ar y fee grwyngau ac yna'n ddweudio ar mas ydw i ddweudio'r tanllach ychydig yn fawr i'r ffordd, a dyna'r gweithio mewn gwahanol ffysigol, sy'n gwybod i ni'n fawr i'r gweithio'r ddechrau. Rwy'n fawr i'r gwahaniaethau, ond rwy'n gweithio'r ddweud, mae'r fawr i'r ddweud yn fawr i'r ddweud o ddweud o'r ddweud o'r gwahaniaethau, yn fawr i'r ddweud o'r ffwg ysgol, ymgyrch i'r ffwyllt yn fwyllt yn fwyllt yn fwyllt yn fwyllt. Ond yna'r anodd yma'r anodd lle i'ch gweithio'r dweud. Ond now I want to move to topology, because that's the subject of these lectures. And I also want to now specify a little bit more what I'm talking about in terms of electrons moving in periodic potentials and so therefore block states of energy bands. So at the moment this whole derivation was just for gyda'r set genericol o bwysig. Ond yna, dyma'r ffordd, o'n... O, mae'n rhaid i'n mynd i'n ffordd yn ffordd y cyfrifol o'r unigion bwysig o Igen States. Ond yn y wneud, ychydig o'r lefio, yn ymdweud yw'n ddweud o'r teulu o'r unigion bwysig. Yn ymdweud, yr unigion bwysig. Yn ymdweud, mae'n ffordd o'n bwysig o'r unigion bwysig o'r unigion bwysig o'r unigion bwysig. Ac ydych chi'n gwneud y bwysig y bwysig o'r ffordd y bwysig yma, dyna i'r bwysig y bwysig o'r bwysig o'r band ymlaen, a chi'n rhaid i'n gwneud hynny. A'r ddigon ni'n gwybod i'r bwysig, y band ymlaen efo'r ffordd o'r ffordd o'r ffordd. A dyna'n cael bod yn y ffordd o'r ffordd o'r bwysig, o'r bwysig o'r bwysig o'r bwysig. Dwi'n ddim yn ymddangos yma, o'r gwaith o'r generysau Then, the berry curvature is no longer this vector field, but actually becomes a matrix, and can be a bit more complicated, so I'm not going to do that in these lectures. So this is for instance a berry curvature of say this lowest band. And so you can see that this is geometrical in the sense that it's telling us about how those eigenstates are changing as a function of the bloc nementor in the Brillawun zone. A frieiniad, y cwm am y fawr a'r ystodio gyntaf, neu y cyfan y tuplodigau yn gywir hefyd, rydybyladiau rhoi byddai'n gwneud y brillawyr yn c closeriad y bwysig. A mae'r cwm yn cael eu cyfeirio'r cyfrif fynd i bod yn cyfyaith bwysig ar y cwm i fod y cyfrifunol. A gennym rin域 am ddigon i dda yn cael ei bwysig y brillawyr hyfodol I wneud y sefydlu o'r f goodsio ar y brylu yn ei ysgwrdd. Felly, mae'r pwysig yn bach i ddweud eu ddweud, ond mae'r sefydlu o'r ddweud, a'r ddweud i'r ddweud i ddweud a'r ddweud i ddweud. Felly, ei ei wneud yno eich paromethau openingr sydd ymddirarchol. Rydych yn ddweud eich paromethau openingr yw yn cynnal o holl49rwyll yn ddweud o'u gyfan'r bobl. Cyrwb ar gyfer y cyffredinol a'i i wneud y ffwrdd gawsbwnae, a oherwydd o'r hyn o'r cyffredinol gyffredinol, oherwydd o'r cyffredinol cyffredinol a oherwydd o'r cyffredinol, oherwydd o'r cyffredinol, oherwydd o'r cyffredinol a oherwydd o'r cyffredinol. A ond oedd gwynnu i ddwy, i ddim yn oedd gwych i gynnwysio'r cyffredinol o'r gael briliwyn yng Nghymru i'r bwysig o'r bandhau energiol. Gwysig ymddiw'r hwnnw i'r ddweud o'r ddweud o'r ffawr i'r cyffredinol o'r ddweud o'r bwysig o'r bwysig o'r gael briliwyn. Felly, yna'n ddim yn ddweud o'r contwyr o'r bwysig o'r gael briliwyn. A yna'n ddweud o'r contwyr, yna'n ddim yn ddim yn dweud o'r bwysig o'r bwysig o'r bwysig. I ddim yn ddim yn ddweud o'r bwysig o'r bwysig o'r bwysig o'r gael briliwyn. Felly, dyna'n ddim yn ddweud o'r bwysig o'r bwysig. Ond ond oedd yma'r bwysig o bwysig?wn ni o'r bysig o'r bandhau periodig i briliwyn. Rwyf i gyfnod i'r bwysig o bwysig o'r bwysig o bwysig o'r lwyddiol i'r bwysig o ddweud o bwysig o'r bwysig o'r bwyysig. ac felly bydd cydonbonell air ar gyfer sefyllwch ar gyfer fy gydag. Felly mae'r eich bod yn cymddiad wych yn上面, mae'r cymdeithiad ar y cyfrifol yn gorffwyd, a'r bod yn cyfrifol yr allan, ond mae'r contwr mae'r cyfrifol yna, mae'r cyfrifol yn y same bobl yng Nghymylio ar y gyrfa yng nghylch. guestsudant mae'r cwrs yng Nghymrydd yn unig. Mae'r cwrs yng Nghymrydd a ddysgu eich gwrs wedi'i eich swydd yr oedd yn olifennig ar y cyfrifol. Yn gelf feddyliau nesaf, dw ni'n mynd i'n glennu'r rhaid i gael ar hyn sy'n mynd i ddweud rhai'r ysgriff fel llunydd a gael o yn cael gweld i'r gwahol, ac yn hollu y ffrifio cael gwag ym mwyliadol, mae'r perthynai ffazfa oedd yn eu cymdeiligau i fath o ffannu, ac mae'r wrthynai'r credu is now the whole Brillouin zone, and that integral over the whole Brillouin zone is now quantised as an integer. And that is what we call the chair number. Okay, so I've actually given a slightly different argument, which is a bit more precise in the online notes, but I didn't want to go through that now, because I think this is a little bit easier to see, but do come and ask me later if you want to know about that. And that is just a simple argument as to why integrating very curvature over this closed surface gives us an integer. And the fact that it's an integer is key because an integer cannot be changed by small deformations. The only way you can change the integer is by doing something so drastic to the system, in this case, closing the band gap and then reopening the band gap to change the chair number. And that's what we mean by the robustness of this topological invariant. And you can also see this is a global property now, because we had to integrate the local Berry curvature over the whole Brillouin zone. Now, I also want to emphasise, because I think this was something that a few people asked me questions about after the lecture yesterday, that these pictures I'm drawing are, of course, not so easy to visualise, because I am drawing this, but what I actually mean when you do the integral over the shaded area is integrating this Berry curvature, which, of course, is a very complicated object. So I'm not talking about seeing the fact that the Brillouin zone is a torus. I'm talking about the topology of the eigenstates that are defined in the Brillouin zone. OK? So that's a key point to have in mind. We're talking about topological properties associated with the eigenstates of a band. And now, why should we care? Why does this have any physical meaning? Well, as I talked about yesterday, the chair number is the topological invariant that underlies the quantisation of conductance in the quantum Hall effect. And this is a very simple argument that requires you to accept a couple of things, but that gives you that idea of the topological nature of the quantum Hall effect quite directly. And in particular, the thing I want you to accept is that if we were to consider a wave packet moving in a geometrical energy band, that the effect of the Berry curvature would be to give this wave packet, so this is the centre of mass position, the centre of mass momentum, and the effect of the Berry curvature is to give the centre of mass velocity this additional contribution, this rate of change of the centre of mass momentum crossed with the Berry curvature at the centre of mass of the wave packet. And the reason I want you to accept this is because I've said the Berry curvature is analogous to a magnetic field. But now I have a Berry curvature in momentum space, so it's analogous to a magnetic field in momentum space. And this is just like a Lorentz force, but with the positions of momentum, the roles of position and momentum switched. OK, this is obviously not a derivation. For the derivation, you can go back to all of these beautiful papers, and indeed the idea comes through already in the early 1950s as band theory was being developed. So this is a property of a wave packet moving in a periodic potential where the eigenstates have got geometrical properties that we have this additional term. So if instead we had the Lorentz force, the Lorentz force would of course be entering as an additional force alongside this electric field. So we've just switched. Now we can ask what happens if instead of a wave packet we have a band insulator. So what does that mean? Well, in this hand-wavy way, we can think of summing up the contribution of all of the different momenta in the Brillouin zone over all of the bands that we have fully occupied and seeing what current we get flowing as a result. So in equations what we can do is we can take this centre of mass velocity, and this is defined with respect to the centre of mass momentum. We can substitute the bottom expression into the top expression, and we can integrate this over all the different momenta within a band, and we can sum it over the number of bands that we have. And we multiply it as well by minus e because we're calculating the current that is flowing as a result, the current density that's flowing as a result of that, and we see what we get. And the first term is a group velocity term, which is just something that most of you will have found in solid state physics, and that depends on the gradient of the band. But the thing is because we have a periodic band, everything that goes up must come down. So if you integrate this over the whole band, the going up and the coming down cancels out, and this term doesn't contribute to a filled band transport. This term on the other hand, after putting in for instance a particular orientation of this electric field, you can show that you get for a field of EY, an electric field EY, you will get a current along X, and it depends now on this integral over the Berry curvature, over the whole Brillouin zone, and that's the Cher number. So this is a very hand wavy derivation based on semi-classics, and actually of course the much better derivation based on the CUMO formula was one of the things that Fowlers won the Nobel Prize last year for. So I do recommend for you to go and read that paper if you're interested, because it's very readable. So that's just trying to give you a little bit of a motivation for why the Cher number should have this remarkable transport properties. And this is just saying words, so we now say that the Cher number is therefore responsible for a quantised conductance, that's just performing this integral, replacing this with the 1 over 2 pi as the Cher number, and the conductance of course JX over EY in this orientation. And I also mentioned that we had a boundary correspondence coming through, which I'm not proving, but which is saying that for every topological invariant associated with the band, we can ask what is happening on the boundary of the system. And in particular I gave you an argument yesterday as to why at the boundary of the system we should have gapless modes because the topology has to change. And in the case of the quantum Hall effect, those are chiral gapless modes. So this is the number of current carrying edge states that we have around the system. And I also said to you that this is a state of matter that comes in the first row of that topological classification, which is the classification for time reversal symmetry breaking. So what do we mean by time reversal symmetry breaking? Well the easiest example to think about is okay we've applied a magnetic field and that magnetic field as an external magnetic field is breaking the time reversal symmetry. This sounds great, we can think of the Lorentz force. Actually I wouldn't be standing here and talking to you if it were really as straightforward as that in for instance cold atoms and photonics where we no longer have charged particles to play with. So we can't just apply a magnetic field and look for a quantum Hall effect. So we actually have to come up with many many sophisticated ways of breaking time reversal symmetry in order to see this and that's going to be a lot of what I talk about in the next couple of lectures. So other ways that we can mimic this physics. And just to highlight some of the models that are really important. So of course if you just think about continuum charged particle in a magnetic field that will give you the quantum Hall effect and that's just the physics of Landau levels. But actually it turns out one of the more interesting models for the purposes of what I'm going to be talking about is the Harper Hosh data model which is a example of what happens when we try to take that quantum Hall system the particle in a magnetic field and put it on a lattice. And so what I want to do next is actually just very briefly introduce to you what the Harper Hosh data model looks like because it's one of the main models that we care about simulating in cold atoms and photonics. And it's got a lot of richness in its structure. So to talk about the Harper Hosh data model I should first talk about how do we include the effects of a magnetic field when we're talking about particles on a tight binding lattice. So you may or may not have seen this before but what we do is we use the piles substitution. Which is basically to say normally we have a tight binding hopping process that takes us from this lattice site to this lattice site. So we have the annihilation of the particle on this site and the creation of the particle on this site with a hopping amplitude JX. But now in the presence of a magnetic field what should happen to this hopping amplitude is that it becomes complex. And in particular it gains a complex phase called the piles phase which depends on the line integral of the magnetic vector potential that the particle has experienced as it did this hop from one site to the next site. And one of the ways the so I've got two indices because I'm now going to do two dimensions. So this at the moment is just X. Because I want to justify this as being like the Aronof bone phase that you get when you hop around a placket. So let's take this case of a particle hopping in 2D on a placket. So this is just a foresight system and if we were to have a magnetic field then we know in quantum mechanics a particle hopping around this closed loop. So if it goes from here to here to here to here and back it should gain the phase associated with the magnetic flux that it's hopping around contained. And that's exactly what the piles phase is designed to do. So for instance we have to choose a particular gauge for the magnetic vector potential because it's a gauge dependent quantity as I said yesterday. And a typical gauge to choose is the landar gauge because it means that we have a magnetic vector potential with only one nonzero component. And so if you look at these these are the phases that you gain as you hop. So for instance as you go from here to here you'll get in my notation from here this theta X M N. As you go up you will get this theta Y M plus 1 N and then this hop I associate instead with net minus this hop which gives you this one and then the final part gives you this one. And in this landar gauge you can see that because I have this dot product of the magnetic vector potential and the link the only contributions I will get will be from the Y links. This one and this one. And if you just put this in you will see that you get this expression which simplifies down indeed to give you the amount of flux that you have encapsulated. The iron off bone phase associated with the flux. So when we write the Harper Hosh data model we normally write it like this. So this is in as I say in a particular gauge where now I only have these piles phases on some of the links. But if I were to do a closed hop around this placket I get the right phase. So this is basically the most natural way of taking a charge particle in a magnetic field and putting it on a square lattice. And one of the amazing things that happens because we have the lattice there we don't just get the continuum physics but we actually get remarkable frustration effects coming about because we now have two length scales in the system. We now have the lattice spacing and the magnetic length. And as these two length scales are commensurate or incommensurate we get the emergence of this amazing fractal energy spectrum. So this is as a function of the number of flux quanta through a placket what the energy spectrum looks like and you can see it has this gorgeous so called butterfly structure. And one of the reasons that people in my field really love this is because the energy bands associated with this have got non-trivial topology and can have different values for the chair number. So I actually showed you an example of this last lecture where the example I gave was actually the Hosh data model for a particular value of the flux. There's loads to say about this I'm afraid I don't have time but if you want to know some of the physics of the Hosh data model then do come and ask me afterwards because it's one of the things I work a lot on. But I just want to say here that from that model that I showed you on the previous slide let's just get it back up too fast. This model this is a model that we know how to diagonalize we can diagonalize it for instance on a computer you can find the eigen states you can calculate the Berry curvature. You can sum up or integrate the Berry curvature over the Brillouin zone and you will find the chair numbers associated with the bands and so they can be everything from like one or one to even like minus four you can have infinite different values of the chair number according to the flux in your system. And it depends sensitively on this value of alpha. I also just wanted to highlight here what I was talking about before with the bulk boundary correspondence. So what I did here was I didn't just diagonalize it taking real space system. I first put periodic boundary conditions and Fourier transformed along that direction and that meant that I could keep one of the momenta as a good quantum number. So on this cylinder what my edge states now look like so my edge states now are states that live on the edges of the cylinder and one edge states in the unfolded kind of open system with open boundary conditions on all sides. Now in this picture gives me these two edge states so I have this part of the edge state which is on one end of the cylinder and the other part of the same edge state which lives on the other side of the cylinder. And then in this gap if I sum up the chair numbers below this gap that's one plus one I should have two and indeed this pair is one edge state and this pair is the other edge state. So this is what I mean by the bulk boundary correspondence. The chair numbers tell me about the number of these edge states and these are chiral because they all go across the gap with a particular group velocity such that I get that chiral transport. Okay so one of the reasons that we like the quantum Hall effect is those edge states those quantized that quantized transport but also because if you do add strong interactions that you get really interesting physics coming out of it. Okay and this in itself would probably be a set of four lectures so I'm afraid I just have a couple of slides about fractional quantum Hall effect. Just to say what are the things that have really got as excited as a field about going here. So in particular this is now the emergence of plateaus not at integers but at fractions of an integer of a filling fraction. And here this is not something that I can explain to you topological band theory because strong interactions are really key this is strong correlations coming through. And that allows for several different things to happen and one of the things that people are very excited about is the possibility that some fractional quantum Hall states. Not all just some fractional quantum Hall states may have quasi particle excitations which can have non-Abelian anionic statistics. Okay so just to say again I'm not saying very much more but these lecture notes are really excellent about the fractional quantum Hall effect so do go and have a look at those if you want to find out more. No I'm afraid not partly because experiments in cold atoms and photonics are not getting there yet. If we manage to do this more practically then yes of a fractional yeah exactly the topological degeneracy is very well defined. Well we have ideas about how to try and actually do this in a photonic system so maybe we'll try and get that done and then I can come back and give a lecture about realising it. Yeah no that's a really important question so this is the question of this is electrons so firstly electrons are fermions and secondly they interact through by the Coulomb interaction and so that's obviously important if I'm talking about a system which has strong interactions playing a key role and indeed there is substantial evidence that a lot of this physics can carry over when instead of Coulomb interactions we have other types of interactions such as contact interactions like we could have in coal gases so you could then imagine having a gas of fermions with contact interactions that should show fractional quantum Hall physics. With bosons you can also get fractional quantum Hall states but they become very qualitatively different so for instance all the filling fractions at which you observe them then become quite different. So for instance the most typical fractional quantum Hall state for electrons that people talk about is like the one-thirds Loughlin state whereas with bosons it becomes the one-half Loughlin state. So it's there's very close analogies but a lot of the details do start changing but we can still do fractional quantum Hall with bosons and with contact interactions and things other types of interactions like that. Okay I did just want to say a couple of words about anionic statistics just to motivate why we care about this. Do forgive me because this is not one of the bits that I'm an expert in but I will try and give you a small introduction and so this is the idea that in two dimensions we can have interesting statistics beyond just bosonic and fermionic statistics and in particular this comes about because we can think that there is a well-defined orientation with which we can exchange particles in two dimensions. So in particular when we exchange particles we can exchange them in a clockwise or a counterclockwise way. Okay so let's exchange two particles and they're indistinguishable so once I've exchanged them I basically come back to the same state but let's say for argument that I could have gained up to a phase factor. This is assuming they're in the same state but in quantum mechanics I could have got a phase factor. Now in 2D we have two options for what we do now. So this is basically option one we exchange them counterclockwise and then we exchange them clockwise. So you can see in terms of the length of my arm is meant to be representing the world line of these particles so as they evolve in time we've exchanged them counterclockwise and then clockwise and then they continue on. The other option so that gains a phase and then undoes the phase so we get back to exactly the same state. There's another option and the key point is that this is different in two dimensions because we can exchange them counterclockwise and then counterclockwise again and I end up with my arms twisted twice like this and this is not the same as doing that in not theory and mathematically speaking. And indeed we can have this phase factor which now can be something other than just pi which would have given us fermions or zero which would have given us bosons and this is because we're in two dimensions. Now this is actually abelian anions and so many fractional quantum whole states may host abelian anions but non abelian anions go a step further because in that case we have some degeneracy of these excitations such that we have a particular state within this degenerate subspace and then exchanging the particles can allow us to rotate within that degenerates subspace so it can change the quantum state. So now we can have this matrix kind of this rotation of the eigen states within this subspace and matrices don't necessarily commute and if they don't necessarily commute then if they don't commute we call these non abelian anions because then we have systems where which particles we exchange it depends now the quantum state that we get depends on the order in which we chose to exchange. So if you have say four particles and you swap one and two and then two and three that would be different to if you'd swapped two and three and then one and two. And people have thought about how this could be used for something called topological quantum computing where the idea is that you take these non abelian anions and you take for instance two non abelian anions and treat them as a qubit and then you make these particle swaps these exchanges which gives you these world lines which are then not something that you can disentangle easily. So these are robust towards perturbations. And so the idea is that how you braid and exchange these non abelian anions could be the execution of logic gates in a quantum computer. This is a very good review which talks about this in a lot more detail. I'm afraid I don't have time to go into it anymore. We'll come back to this a little bit later. But the idea that we're trying to get towards is okay if we have fractional quantum whole states they could have these really cool quasi particle excitations therefore let's try and make fractional quantum whole states. And we're still at the stage of trying to make the fractional quantum whole states in my field. Okay so very quickly before I get to the last part I just want to say very few words about some other aspects of quantum whole systems. So as I'm quite short on time I'm going to go fast here but do ask me more questions later. The first thing I want to say is that I showed you this topological classification with many other dimensions other than just the three spatial dimensions we normally live in. And that's because mathematically you can calculate what the possible topological phases could be. Okay can you realise this in an experiment? What I will try and persuade you about tomorrow is that yes you can using some tricks. And so therefore we can care about these topological phases that exist in dimensions higher than three. And in particular one of the things I've worked personally on lot on is this four dimensional quantum whole effect which here written in terms of the conductance, sorry the current density due to an electric field you can see is now a non-linear effect so it also requires another magnetic perturbation but now it depends on a 4D topological invariant that's called the second chair number. So do come and ask me more about that but basically it's just saying that there are generalisations of that chair number to higher dimensions that lead to new quantum whole effects in 4D, 60 and 80. Another thing I just want to mention very briefly is the idea of a topological pump. So this is really a big idea in the fields of cold atoms and photonics at the moment which is that okay we have a chair number and to get the chair number we integrated a very curvature that depended on kx and ky. But did they really need to be kx and ky? They just need to be periodic functions and in particular you could imagine having a very curvature that was defined with respect to kx and a very curvature that was also a function of a periodic parameter, now a periodic pump parameter that is something we're changing over time in the system. So this is now a dynamical system but we're changing a system dynamically and periodically such that the eigenstates have a very curvature with respect to kx and this pump parameter phi and then we can again talk about chair numbers but now we're actually one dimension down we're in 1D and this would have physical consequences in the centre of mass shift for instance after every period. So this is a kind of a quantum version of an Archimedes screw that we're turning this we're modulating the system periodically and we're getting the transport of particles through the system and it goes right the way back to thawless in the 1980s just after the discovery of the quantum Hall effect. So it's basically just saying okay this doesn't really need to be two dimensions it could be 1D plus a periodic dimension. However of course we have to be careful because this isn't really a dynamical variable this is something fixed that we're doing to the system. Okay I think I'm going to skip the next bit but this is just talking about how we can go from a 2D quantum Hall effect to a 1D topological pump and this is just the recipe that you need to follow in order to get there. So it's basically Fourier transforming with respect to one of the dimensions and then replacing that Fourier transform dimension with an artificial pump parameter. So it's what we call dimensional reduction it takes you from a 2D topological system to a 1D pump and the important thing to note is that this is much much simpler because we started with the 2D Hofstadter model which is a charged particle with gauge fields and all the rest of it and we ended up with a dynamical 1D hopping model just with on-site potentials. So we've actually really decreased the complexity of what we need to simulate. Okay so now I want to say a few words about some of the other topological phases within this periodic table. So first off we have these things called topological insulators and they are what really started the system going in 2005. And I want to highlight firstly one of the things about time reversal symmetry which is the key to understanding these phases of matter. So these unlike churn insulators have time reversal symmetry and the key point here is that they have time reversal symmetry for spin a half particles. So time reversal symmetry for spin this particle is actually just charged conjugation because you can think about that momentum has to flip. Momentum in quantum mechanics is IH bar grad. So flipping it you can just change the sign of the I charge conjugation. If you want to do spin half you also need to flip the sign of the spin in time reversal. So time reversal also needs to flip the sign of the spin so you need to add in some Pauli matrices there to do that flip. Now these symmetries have very different properties if you calculate what happens when you take that symmetry operation and you square it so you apply it twice. And in particular in the first case you can show that T squared is just plus one because charged conjugation and then charged conjugation again just brings you back just plus one the identity but T squared in the second case is minus one. So this is actually what I meant in that topological classification when I had those columns about the symmetry. The idea that T the time reversal operator T squared could be plus one minus one or zero if it was broken. And this is a general thing that distinguishes bosons and fermions. And in particular one of the important things for fermions is that T squared equals minus one which means that we have something called Cramer's theorem. So Cramer's theorem is asking ourselves what happens when we have a system with time reversal symmetry. So let's have a Hamiltonian with some eigenstates. And this Hamiltonian has got time reversal symmetry which means that if we apply the time reversal operation like this that we get back to the same Hamiltonian. Now what if we take the time reversal operator and act it on a state and then act the Hamiltonian from the right hand side. Now I can act on the left hand side with T, T minus one because that's just an identity, it's T times its inverse. But this part here is just H. So now I have H acting on Psi which gives me E. So now E is just a number so I can take that through, it's real. So I can take that through T and then I have E T Psi. So H T Psi equals E T Psi. So that means T Psi is also an eigenstate. So that is fine if I have a time reversal symmetric Hamiltonian then T Psi is also an eigenstate of that Hamiltonian. Now the question comes is this the same state? Have I just operated with time reversal and got the same state? So if it were the same state then T Psi should just be Psi up to a phase factor. Now let's see, for fermions T squared equals minus one. So let's start from minus Psi because this is slightly easier. So minus Psi, well minus one so that's just T squared Psi and then I act with T Psi, one of the T Psi's to get me E to the I Psi. Then I take E to the I Psi, E to the I alpha through that T but T can change charge conjugation so it flips the sign. So we know that T has got charge conjugation in it so you get that. And then T acts on the Psi again and we get E to the I alpha. Now those two go away so this tells us that minus Psi equals Psi. Ah no that's not good. So this was if they were the same state then we get this contradiction and that tells us for time reversal fermionic systems where T squared equals minus one all the eigenstates have to be twofold degenerate so Psi and T Psi have to be different states. Good? Key point for bosons T squared equals plus one so we don't have Cremus Theorem and actually this caused a lot of problems in photonics for a while before we figured this all out. But I'll talk more about that tomorrow. And so what they realized in 2005 was that this has really important consequences when we're talking about topology because if we now consider a Hamiltonian H of K then the time reversal operation T H of K T minus one so it's H of minus K but then if I look at what's therefore T acting on Psi should do it should flip the Psi to minus Psi. So these are time reversal symmetric partners and they have to have the same energy because I said that Psi and T Psi are both eigenstates with the same energy. So this tells us that there's this symmetry in the Brillouin zone. That's cool but what about those quantum Hall states? Well now you can see straight away that a quantum Hall state has to break time reversal symmetry because it should have a reflection symmetry, the dispersion should have a reflection symmetry about this point and it doesn't. Okay? The chiral edge states on their own are not possible with time reversal symmetry for fermions. And so what they asked was well what if we just take a quantum Hall state for one spin and the opposite quantum Hall state so would the opposite say magnetic field for the other spin and add the two together and then we have a chiral edge state going in one direction from say the spin going up going this way and a spin down going this way and that now does satisfy this required symmetry, the time reversal symmetry. And the key point is that here these bands are now made up of these time reversal symmetric partners so they're protected by Cremus theorem. It means we can't open this so at this point we can't break this open, we can't mix the states and so these have to be very robust. And what they found was that this is the example of a time reversal invariant or quantum spin Hall system. So in particular we have two possibilities. We have that the points at zero and points at pi have to be the Cremus degenerate points and either at the Fermi level we have an even number of pair edge states crossing this in which case we can imagine taking this whole band and pushing it down out of the band gap perturbing it away or we have an odd number in which case this is robust because we can't, we don't have the freedom anymore to push all of the edge states using perturbations and this gives you, sorry I'm going very fast but this gives you the idea of this topological invariant. So this is just saying in words that we can imagine taking spin up plus phi and spin down minus phi and then in the simple case if we have spin conservation that gives us the topological invariant very straightforwardly by just counting how many edge states that we have and this is the picture that we have in mind. And now we could include spin mixing terms but they actually preserve the Z2 classification provided those spin mixing terms are time reversal invariant. Okay so take home message, this was quite complicated but quantum spin hall system at the basic level you can just take copy of quantum hall for spin up opposite copy of quantum hall for spin down add the two together and that's already a pretty good quantum spin hall state from what I care about. But all of the other things you can do to the system lead to the whole plethora of interesting phenomena in topological insulators and really revitalize this field. Okay I haven't explained a 3D topological insulator because it's a little bit more complicated but I quite like this picture of it so as opposed to having these edge states this one. So these are now surface states that are going across and so there, this is an example of how the spin surface states would cover the surface. So you can use, you can relate again everything comes down with topological insulators to the time reversal symmetric points within the Brillouin zone. And in particular in the 3D topological insulator case it's about how the Fermi surface at the surfaces of these 3D materials have a 2D Brillouin zone and in the 2D Brillouin zone the Fermi surface either encapsulates one of these time reversal points or I think it's an odd number or an even number of these time reversal points is then related to that. So yes an odd number so it's about in the surface Brillouin zones yes in these time reversal and variant points where the Fermi surface is going and according to that you can derive the strong topological invariance which is the one that is most important for characterizing 3D topological insulators. If you add interactions then I think there is some thought that the topological, the fractional topological insulators would then be quite related to a fractional quantum Hall effect and I think they should have topological degeneracy but I really can't say any more about that. I'm not so much of an expert. Okay right so I think I will choose to skip the SSH model. I'm afraid I'm running out of time sufficiently much but this is just you can ask me about it later if you like. This is a class that has now this mysterious chiral symmetry and we like it because it's actually a very very simple example of a topological phase. I won't go through the slide but just to say that it's the model for polyacetylene and it can be thought of as edge states that arise when you have a tight binding lattice with two different hoppings and depending on the value of the two different hoppings this bipartite lattice can either have protected edge states at the boundary or not. I want to say something very quickly just about topological superconductors before the chair tells me to sit down and so therefore let's just have a quick look through these slides. Now the point I want to make is that I've talked up to now about really independent electrons but all that I really needed was the fact that I had gapped bands in a system and so I could think about another case where I have gapped bands, gaps in my system which is above a superconducting ground state. So this is just a mean field superconductor so in the mean field we've replaced some of the interaction with the mean field pairing potential and this is just the Boglubov-Degen Hamiltonian and this is usually gapped according to the superconducting pairing potential but one of the remarkable things that people realise is that in a topological superconductor it can be protected states within this gap. So to see that we can just think about particle-hole symmetry so this is the other generic symmetry that I mentioned and now particle-hole symmetry is very much like the time reversal symmetry except that there's now a minus sign here and this means that if we were to go through the same sort of derivation that I had before what we would now find is that particle-hole symmetry means that every state psi has a partner now not at e but at minus e because of that minus sign so particle-hole symmetry gives you a reflection symmetry now around equal zero. So before time reversal gave us a symmetry around with respect to k equals zero but now we have a symmetry with respect to equal zero and also k going to minus k but we also have the idea that we have a particle-hole redundancy because actually we know that creating a quasi-particle in this site e is actually the same as removing a quasi-particle from this state minus e because that's what we have in a superconductor and so that means that now we can imagine the case where say we do have states within a band gap of a topological superconductor so normally this gap is delta the mean field pairing potential but now we have two possibilities at say in 1D we can have states at the edge now if these states are away from e equals zero then you're able just to perturb them out of the gap because as long as the energy here and the energy is minus the energy here you can just perturb those out of the gap but you can't if you have a state at e equals zero a single state and this state at e equals zero because of this redundancy I told you about actually also has the property that it is the creation operator for this state is also the same as the annihilation operator and that is what we call a myer on a fermion so this is just saying that if we have particle-hole symmetry it's possible to have states that are pinned at e equals zero and that are topological and that are these myer on a fermions and in two dimensions myer on a fermions we can think about their exchange statistics as another type of non-abelian anions so people also like this because of its links to topological quantum computing now in the notes that you can find online I give you the example of one of the simplest topological superconductor models which is the QITAF chain and this is very much like the SSH chain but now we don't have different hoppings but we have this pairing potential and so you can go through that slide to see how we get the appearance of e equals zero edge states and this is just to say in two dimensions we can have 2D topological superconductors where now the topological invariant tells us about the number of myer on a edge modes around the system and in particular if we have a vortex in the system then we can think about a vortex in a superconductor as creating an extra little edge for us so we can end up having myer on a modes these edge modes hosted also at that vortex and by tuning a flux through that vortex we can make that have zero energy and so the idea in a topological quantum computer based on topological superconductors would be to have myer on us in these vortices and then to move the vortices around and then braid them in order to do that braiding I talked about to give you the logic gates and just to say so one thing that I want to emphasise is that these are quite unusual superconductors cos they're not S wave pairing normally in a superconductor you're pairing spin up and spin down but here we're actually the simplest models or spinless pairing and so we can get this for instance with various types of P wave pairing which means that we have to try a little bit harder to find them in the laboratory and so people have been looking and there's hope that maybe we could find them in some types of unconventional superconductors also the quasi particles that I mentioned some of them are examples of myer on a fermions or in proximity devices so this is where we have for instance a system with strong spin orbit coupling coupled to a superconductor and the idea is now that the spin orbit coupling changes the dispersion so it means that you have spin up dispersion going one way spin down dispersion going another way a momentum space and if we open a gap for the Zeeman field so this is the spin orbit coupling plus a Zeeman field we now can have a fermi energy that is intersecting this point and this point and so you could have even S wave pairing giving rise to an effective spinless superconductivity and there is mounting experimental evidence that indeed in this kind of system you can see myer on a fermions so that was a bit of a whistle stop tour through some of the more exotic topological phases so I spent most time on Quantum Hall because that's where cold atoms and photons really is at the moment and I'll be telling you about that tomorrow but I also wanted to wet your appetite for some of the more exciting things that we have on the horizon and in particular towards this topological superconductivity which solid state physicists are still struggling with so maybe if we can improve the cold atom set-ups sufficiently we can get there first thank you