 Hynny'n meddwl hwn. dyma'r iawn hyd, a gwnaeth beth sy'n gobeithio'r organiwyr i fy� ar ai'r gwneud yma ac yn gyflym iawn. Byddwn i installio'r bwysig gydaeth yn ddim. Gwyddoedd mynd i'r ffrindiau sy'n gobeithio gen i gwirio hwn neu'r bwysig o'r hwnnw yn gyflym iawn i gwirio hwnn wrongfyrdd wath Đogol, ac mae hynny'n broses'u ei ddysgol sy'n gweithio'r bwysig. A fyddwn ni'n gweld bod yn cael ei gwrtho ddaeth Something that's a little bit outside of perhaps the main thrust of this school, so I'm not really talking about quantum information or those sorts of things, but another aspect of modern advanced quantum science which we hope is also taking us on a path towards some future quantum technologies. So the organisers have asked that I divide up my lectures in order to first give you a bit of an introduction to topological phases of matter fel rai o beroedd yn cael ei dynol i'r fullfynol. Bydd yma'r dduodau y lŵn hwn o'r olde company o'r pryd yn gwirion a hwnnw o'r profiad ond yng nghyddedig yr ystod o'r ffazor diolch, yr hyn o'regoedd diolch o'r ffazor diolch, nad oedd yma'n gêmio peaniaethau a fyddwn ni'n adeiladru o'r modd o'r teisio ddwygr yng nghymru ac yddechisio ddisglwyddol o'r ffazor diolch o'r ystod o'r lles. ac yn eich clywed o'r lech yma ym mhwy o'r ffaith mewn cymryd o'r tynnu hwnnw, i ddim yn ymgyrchu ddaeth ffasiau yn cymryd o gael ymweld a'r ymweld i ddweud o'r ddweud o'r ffaith. Mae wedi bod yn bwysig o'r ddweud o'r ddweud o bobl yn gweithio i ddweud o'r ddweud o'r ddweud o'r ddweud o'r ffaith, a how this can help us to learn more about these topological phases of matter, but also have some quite interesting possible applications and reach some new regimes that we haven't been able to see with just solid state materials. Okay, so, as with all the lectures today, do feel free to ask questions as we go through. We also, I think, have the question time at the end of the afternoon so do come and ask more then. ac wneud ei ddweud i wneud y llweddau'r gwahanol iawn. So mae llweddau yn siaradな nhw'n ffordd. Ar hyn arwm y gwahanol, mae'r bellấ wneud, ac mae'r bwyddon yn ôl i yw'r adnod i'r ffordd. Byddw'm ar y gwrthod gyda wnaeth, cy hurtsant ddweud y llweddau, ystod y gallwch yn llweddau y gwahanol. Pan ddrwy. D diplomacy'r mawr e'r ffordd o'r ddaw, ac mae'r llyffaith cwyl i'r lluniau ddefnyddio'r llyffaith o'r llyffaith ymlaen, ac rwy'n meddwl i'r llyffaith fy mwy ffawr i'r llyffaith. Rwy'n meddwl, y cyfan ni'r llyffaith gyda phasig allanol â'r phasig a'r materau'r ffordd yn hwnnw fel gwahanol. Ac mae'r llyffaith gwahanol yn hwnnw yn unrhyw gwych ar yr ysgrifennu, ac mae'r llyffaith yn ymddangos i'r llyffaith gyda David Tanc Cambridge, a gyda chyŵr gwwysig ymweld gweithio'r dynnwch dechrau'r dynnwch hanfawr arall intensiol hanfwyr. Mae eich phasio o popologiaeth yw'r hallig o'r maen nhw. Mae oedd eich drefwyr yn gweld yn fwy o bwysig rwynebu'r byw. Byddwn i'n meddwl, byddwn i'n meddwl agnig ffasio'r furiaethau rwynebu a byddwn i'n meddwl a byddwn i'n meddwl arwnaeth. Ynw, yna dwy o'n meddwl. ac yna ein gwaith bwysigwyr ymlaen i'n un i'w ddod o gweithio'r teulu o ddod o bobloddiogol gwaith. Ac rwyf wedyn bod os yw'n ei bod yn ystod o gweithio gyda'r efrif? Ymlaen i'w beidio'r efrif yma, byddwn i'n meddwl gyrdd yma, mae'r gwneud yn ddod i'n edrych. Mae'n ddod o'i ddod o'r gweithi'n edrych. on-line course that you can do so it has all of the questions and tutorials that is accessible here. So as I say, all of this is available on my slides, on the website so you don't have to write it down, but those are some great references for everything that I'll be talking about in the next two lectures. And then I'll give you more references as we move on to the cold atoms and to the photons. So what is topology? Let's just start with the very basic idea of topology that we get from mathematics. And for instance we can talk about topology as a way to classify different types of surfaces. So this is an example that I'm sure many of you have seen before, which is how can we tell whether an orange is different from a donut. And the idea is that these surfaces, regardless of the local geometrical details, are characterized by a global topological invariant. And in this case that invariant is the genus, it is the number of holes that this surface has. And so this is a little animation that I stole from Wikipedia, which is showing you that as you take this coffee mug and you deform it into a donut, you're keeping the number of holes the same. And that means that this surface has always got the same topological class. It always has a genus equals one, even though you're twisting and smoothly deforming it. So the idea is that in mathematics we have sometimes got these global properties. They are integers, which is one of the reasons they're so robust because you can't change an integer continuously. You have to do something dramatic. You've got to tear a new hole in order to get from one genus to another genus. And these are properties that can be used to classify different surfaces. Now, what about physics? Standing here as physicists, then what really started this idea of can we do these same sorts of topological phases but now with solid state systems was beginning with the discovery, the experimental discovery of the quantum hole effect in the early 80s. And this was discovered by Claes von Klitzing and others and he won the Nobel Prize in 1985 for realising that when you take a cold, quasi-dimensional 2D electron system and you change the magnetic field that you're putting perpendicular to the surface and you're injecting currents and measuring the hole voltage, so the voltage perpendicular to the current that you inject, that you see these amazingly robust plateaus in the hole resistance. So the hole resistance being related to the hole voltage and the current in this way. And these plateaus are incredible because they are quantised very, very robustly and precisely as H over E squared with an integer and this integer N. And this precision is so good that this is becoming our best definition of electrical resistance and H over E squared. That's so robust and it doesn't depend on the details of all the geometry of the sample. If you have a sample with disorder it's still robust and this really posed the mystery of why? How can we get something that is this robust? And it was Thaolus, David Thaolus and others who showed shortly after this experimental discovery that the reason this is such a robust phenomenon is because this is the first example that we know of a topological phase of matter. And you probably heard about this in the news because this was of course part of the Nobel Prize in 2016. So I was very excited about that especially because it meant there were popular science articles that I could give to my parents so that they had some hope of understanding what I do. It didn't really work but at least it gave a bit more incentive to this field and brought a bit more of a highlight. But this is just the first example. What we realised since the early 80s is that topological phases of matter are actually a very, very broad class of systems. And it's actually a whole other way to understand and classify different phases of matter. So the phases of matter as you learn them in undergraduate physics in most universities is that we have this paradigm of spontaneously broken symmetry for instance. So you can think of the difference between a disordered magnet where the spins are pointing everywhere and then as you call it down or as you change other parameters you go through the transition and the spins align and to get a pheromagnet. And so that's an example of in the disordered state you have a local order parameter like the magnetisation which is zero and then as you go through the transition you have that local order parameter taking a nonzero value. So we can say that in that paradigm different phases of matter are characterised by different values of that local order parameter and a spontaneously broken symmetry. Now topological phases of matter are also very distinct phases of matter but do not fit into this paradigm because we don't have local order parameters. We have global topological invariance in analogy to that genus that I was talking about a moment ago and I'll tell you some more about the topological invariance. And we also can't think about the spontaneous symmetry breaking in the same way. We can't think about if we see a system that spontaneously breaks a particular symmetry then that's the topological phase. It doesn't work like that. Symmetry is important but plays quite a different role. So it's another paradigm of phases of matter but one that is generating an awful lot of richness as we'll see through these lectures. Oh have I skipped a slide? No I think that's... Okay yes so why am I standing here? Well from my title I'm not talking about electrons really. I'm actually a photonics and cold atom person so I personally work on topological phases in photons and cold atoms because we have much greater controllability and tunability of those systems. So it's a wonderful playground for engineering different Hamiltonians that we really want to see the topological properties of. And also I'm excited about how we can use the cold atoms and photons to explore new things that weren't possible with just electrons and solids. And this is going to be the focus of lecture three and lecture four. And I also want to say I feel like I should situate myself within the school because it's maybe not so obvious where I fit in. But this is a very exciting area that's getting a lot of attention at the moment. And it's very useful because it's showing us how we can use the cold atoms and photons to simulate, to do quantum simulation of interesting effects. And also there's this hope that maybe some of the stuff will have some practical applications. What is topological in a phase of matter? Okay the short answer is the wave function. We're talking about the topology of the wave function. The more precise answer for these lectures is I'm going to be talking about topological band theory. So that means that I'm really limiting myself to systems where we can talk about the single particle energy bands of a system. So typically we're talking in the first two lectures just think about electrons moving in a solid. So we have electrons in a periodic potential. And through blocks theorem we know that the electron wave function can be decomposed into the plane wave and the periodic block function. And those periodic block functions have got satisfied the block Hamiltonian and the energy bands. So we can plot them in the Brillouin zone in this way. And I'm interested in the topology of those eigen states, those block states here. And how those block states vary over the energy band and what topology we can assign to that. I'll talk more in detail about this of course. But here's just an example of a topological system where with each energy band we can associate an integer. This is called chair number. There are other possibilities. And I'll talk a lot more about chair numbers, so don't worry. But the idea is that these eigen states taken together as an energy band have got some topological properties. Now when is this good? Well this is good for many many topological phases matter that we're interested in. So it's the same. In this particular model this is actually a Harper Hofstad model with flux equals one fourth. So it's actually got a symmetry between the top and the bottom. So the bands look actually the same. So in this case they've got the same topology, the top band and the bottom band. Which is a characteristic just of this model. Okay, so this is also an example of a particular model which will show the quantum Hall effect. So I'll introduce this later. So integer quantum Hall effect can be understood with this independent energy band theory approach. As can topological insulators. So okay I'm using terminology which means that I say topological insulator for time reversal invariant topological phases in a way that I will define later. Some people in the field use topological insulator to mean top everything including quantum Hall systems. But I'm going to be slightly different with that terminology just to warn you. But it should be clear as we go through. And the other thing that we can talk about is strictly speaking when we don't have these are electron energy bands. But when these are quasi particle bands. So if we're talking about the topology of for instance quasi particles in a superconductor or a superfluid. That also fits within this particular paradigm. Now what topological band theory cannot do is it cannot tell you about systems where really strong interactions are important. So for instance fractional quantum Hall physics. We cannot describe just using the single electron picture all the time. And that's very interesting and that's a whole other talk in itself. The reason that I have chosen to focus on the topological band theory is because that's where the fields of cold atoms and photons currently is. We're at the stage in cold atoms and photons that we're really good at getting topological bands that are single particle energy bands. But we can't yet do the strongly correlated topological physics. That's what we'd like to do but we're not there yet. So to help you understand the state of the art this is the way that I'm structuring this talk. So within topological band theory we have bands. But what does it mean to say a band is topological? And how can we say that this band has got a different topology to some other band? Well a very useful concept is that of topological equivalents. Which is saying that we can define two bands as being topologically equivalent. If we can smoothly deform one band into the other band without changing, without closing the band gap. And this is crucial. So this is just a schematic of what I mean by this. This is taking the same model that I showed you before. So it's the Hofstadter model and in this particular case there's a parameter. Don't worry about these details but just think there is some parameter of the Hamiltonian that I'm tuning. And as I tune this I can go from a system which is topologically trivial to a system which is topologically non-trivial. And the only way that this can happen is if at the point where the transition occurs. Which is corresponding to, sorry this projector I don't think has shown. Let me just see. Okay so I'm afraid this is a little bit hard to see but this picture here corresponds to in this regime. This picture here corresponds to exactly at this transition. You can see all this on the slides online so check there if you're not sure. And at this point the gaps close so you can see that here now we have these band touching points. And that is required for the topology to change. Okay and this is part of what we mean by topological robustness. Because anything that we're doing to a band as long as we're not closing that energy gap is preserving the topology of the eigenstates. So that's one of the central ideas and that's one of the reasons why topological physics is so interesting because it's very very robust. Because if you add perturbations, if you add stuff to your system that changes the band structure a little bit provided you don't do something so dramatic that you close the band gap your topology stays the same, any physical quantities that depend on that topology will stay the same. So here for instance I'm talking about the topology of the single band. So if you were to, when you wanted to assign a topological invariant you could assign it to this band on its own or you could assign it to a group of bands and then that topological invariant would not change even if you had band gap closings within the group of bands that you selected. So you can choose, basically you're choosing how much of your spectrum you want to group together. Here I'm grouping together this band. Okay, technically here I have two bands so I've grouped those two bands together and here I can have band gap closings within these two bands but because the topology I'm talking about is as that, as a unit that's okay. Yes, so we'll come on to that. So that is the, okay, this is where the word topology has two meanings. So that means the topology of the parameter space in terms of how the block states are connected to the other block states but then we can talk about the topology of the eigen states themselves so how those eigen states change and that's the topology that I'm assigning this index to here. So in, yes, so in some sense the, because of the period of boundary conditions it's always a torus but how those eigen states are actually winding on that torus is what I'm calling this topological index. I'll show you some more examples later. So band gap is crucial in the sense of the band gap is what is protecting the topology. So when the band gap is going to zero then you will see for instance with the topological properties that I defined that that's the point where you can get a topological transition between two topological states. So as long as the band gap stays open I'm in one topological phase. Was there another question? Yeah. No, that's a very good point. It is not a chance at all. So you can show that the sum of the chair numbers over the whole system have to add up to zero. That is a fundamental property of the chair numbers. Yeah. So that is not a chance at all. It's a symmetry. We can talk more about it later. Okay. So another point that arises from this which is one of the reasons why people love topology is because I've just said that the band gap is crucial. So open band gap can be in a topological phase and then at this band gap closing we can have the bulk topology of the band changing. So here for instance the lowest band has topological index chair number zero and here it is one. And at the point where we get the transition from one to the other the band gap closes. Now you can turn this around. Here I plotted this as a function of the tuning parameter. But what if now I had a topological material next to a vacuum? So an interface between a topological material and a vacuum. Then the vacuum would be like it was on this side and the topological material would be like it was on this side. And what I've just shown you and you would be able to see perfectly if the slide had come out well is that at this point here the band gap closed because we had to have the band gap closing to have the transition. And that means there must be gapless modes confined to boundaries of topological materials. So because between non-topological and topological so topological and the vacuum we have to have the band gap closing means there will always be gapless modes at the surface of a topological material. And we'll see that this is one of the reasons we really love topological physics because those gapless modes can be very, very special. And that's called the bulk boundary correspondence just to say it's much deeper mathematically but I'm not going to go into that. Now I did say that symmetry played a role. So what is the role that symmetry plays? Well it is a guiding principle for us. So we want to be thinking about what kinds of symmetries are possible in our system and that will allow us to talk about what types of topology are possible. What types of topological phases of matter are possible. And so in a moment I'm going to show you the classification table. But firstly I just want to say so we can obviously think about crystals that have certain spatial symmetries. So certain types of reflection or glide symmetries or rotation symmetries and those are great. They give us extra constraints that can also lead to interesting things but those are not very generic. Those are very easy to break. You add a defect and you can easily break a reflection symmetry. So what we tend to do is we tend to ignore these symmetries at least at the lowest level is not being so interesting and focus on the most generic symmetries that we can possibly have in a quantum system. And the most generic symmetries that we can have are time reversal symmetry and particle hole symmetry. So time reversal symmetry you can just think of okay so what does time reversal symmetry do? Well it's like reversing the direction of time. So for instance if we have a Hamiltonian as a function of momenta it should flip the momenta. But the system should remain invariant under this. And this time reversal operator is an anti-unitary operator that reverses that we will define for particular models but that is going to reverse the direction of time in our system. Another generic symmetry that we can have is particle hole symmetry. And so we'll see this particularly for instance systems like superconductors and we have whole classes of topological superconductors that have particle hole symmetry. And so that's saying that the symmetry of having a particle is related to if you can either add an extra particle or you could take out a hole. You could add a hole take out. So then we also have combination of the particle hole and the time reversal symmetry which we call the chiral symmetry. And between them these symmetries are the most generic things that we can think about. And what they have done very recently starting from 2009 is classify what possible topological phases of matter are possible. So here I am not deriving this I'm afraid because deriving this is very, very complicated. But these are references where you can find everything discussed in much more detail. But I want to use this as the structure for the discussion that we're going to have. So we're interested in these kind of generic symmetries and we're going to be talking about different examples from this table. So what really am I talking about here? So first off let's look at the symmetry table. So this is called sometimes the tenfold way because these are the ten possible symmetries that we can have with those time reversal particle hole and chiral. Now you can show that time reversal symmetry operation if you have T being the time reversal operator that T squared has to be plus or minus one. Come and ask me afterwards if you want to know more details about that. But that means for time reversal symmetry we have three options. We can have plus one, zero and minus one. And next lecture I'll show you an example of plus one and minus one. And zero means it's broken. Particle hole we have the same three options because we also have to have that P squared equals plus or minus one. So we have zero plus or minus one here, zero plus or minus one here and then chiral is a product of time reversal and particle hole and that just gives us one extra class on top of the other nine. So these are the generic symmetry classes and then according to the dimensionality this is the table of topological invariance possible for the fermions. And I love this because this to me is a bit like a periodic table is to a chemist. I look at this table to see where am I, what is possible. So if you have for instance a system that has none of these symmetries and you're in dimension three then you look here and you see it's zero and that tells you straight away you can't have a robust topological invariant in that type of system. If instead you're in dimension two you see z and that says you can, you can have an integer topological invariant. And this is just I should say for the non-interacting fermionic Hamiltonians with these types of symmetries. We can also consider adding back in those spatial symmetries. We can also consider classifying defects and we can also say something about gapless systems but I'm not going to say anything about that. And so this is what people kind of are referring to when they're talking about topological phases matter. It's this table and what we can explore within it. Be careful because just because there is an entry in that table that says z or z2 it doesn't mean that you're always going to be in a topological phase when you're in that symmetry class because zero is a perfectly fine topological invariant and zero means it's topologically trivial. So this is why it's very different to the spontaneous symmetry breaking paradigm because just because we're in a particular symmetry class does not mean we are guaranteed to have non-trivial topology. We actually do have to calculate and check and see what our system is doing. So why is the symmetry important at all? Well it tells us where to look. So straight off I can say if you have no symmetries and you're in three dimensions there is nothing topological in a robust sense. You maybe could add some extra reflection symmetry and get something kind of topological but it's not really robust. You can break it really easily. But also it tells us within a particular symmetry class what kinds of things can we do to that system and keep it topological. So if we have a topology that relies on time reversal invariants and we add some magnetic impurities that break time reversal invariants our topology is gone. It's not robust. So this tells us what we can do while keeping that topological class. Now I want to get to what I'm actually going to tell you about. So I can't tell you about all of those classes but I'm going to tell you about some of the most important entries in this table. In particular I'm going to focus the rest of today and probably a bit of next time on the quantum Hall class. So quantum Hall classes are in the first row so they break time reversal invariants. So we saw that already because we were talking about quantum Hall effects of a 2D electron system in a magnetic field and a magnetic field, an external magnetic field breaks time reversal symmetry because it forces electrons to be chiral and that you can't reverse without reversing the sign of the magnetic field. So it's breaking time reversal symmetry. They also don't have any of these other generic symmetries so we're in this class A. And one of the reasons that we love this class so much and I spend so much time on it is because if you want to talk about cool things that you could do applications it's actually kind of the most interesting class because it's really really robust because I've already broken all my symmetries which means all the perturbations I add are going to stay within that symmetry class. Now the other things I do want to talk about are a bit about topological insulators so they're a bit like class A but now with a time reversal symmetry. And this is what really sparked off this explosion in the field in 2005 was the discovery of these topological insulators. Theoretically, the theoretical prediction and understanding of them. As I say, if you add some magnetic impurities you can break these so I don't like them as much but they're very very beautiful. Then there's also in particular some other classes that I'll talk about the SSH model just very briefly and these classes here which you can see have got particle hole symmetry and so that means we're talking about topological superconductors and those are very cool because as we'll see they can have modes in them that are myerana modes that people are interested in interesting non-abelian properties and statistics so that takes us in the direction towards topological quantum computing. Okay, so I apologise for not having derived this table but this is a picture to have in your mind of a plethora of possibilities and now let's dive in and see what a particular example of a topological phase actually means. So the 2D quantum hall effect let's go back to this. So just to recap what I said before we have a system where we can have these plateaus in the hall resistance and this is an integer and this integer we're going to show is related to the sum over the occupied chair numbers of bands so I'm going to define for you what a chair number is I'm going to talk about how we get them and this chair number leads to this robust behaviour and as we said it's very robust because this is a very robust topological class we can add disorder to the system we can do crazy stuff to the system as long as we don't close those band gaps then the band gap at the Fermi energy in this case is the crucial one as long as we don't close that we're going to have the same topological behaviour and also I said something about this bulk boundary correspondence about the gapless modes and one of the reasons we love quantum hall is that the nodes are actually one way propagating chiral edge states in this case which means that at the edge of the system we have an edge state that connects the valence band and the connection band that is only in one direction so as a group velocity that for instance in this case is always positive and connecting the two bands so classically you can understand this because classically if we have these electrons which are charged particles moving in a magnetic field in the bulk of the system they're doing closed cyclotron orbits but at the edge of the system we can see something special can happen because as an electron tries to do a cyclotron orbit it's bouncing off the edge of the system it skips along the bottom and the quantum version of this are these chiral edge states that come through the bulk boundary correspondence so what is a chair number? How do we get to a chair number? Well, I'm going to take a step back and introduce it through something called the berry phase and one of the reasons I've chosen to do this is because this way of understanding the chair number has really inspired a lot of the work in cold atoms and photonics as we'll see in a little bit so what is a berry phase? As we'll see berry phase is very intimately related to these topological properties now we're considering just a Hamiltonian that depends on a set of parameters so R1, R2, R3 it doesn't really matter what they are these could be external fields for instance we could be talking about a spin in a magnetic field and this could be the direction of that magnetic field now what I'm going to consider is I'm going to consider the case where we have we know the normalised non-degenerate eigenstates of this system you can generalise this you can talk about degeneracies but it complicates things too much for what I want to present here so we just have this Hamiltonian and these are the eigenstates N of R now we ask ourselves what happens if we take a state prepared at R at time equal 0 so N of R at 0 and we evolve it under adiabatic variation of the parameters so I just said these states are non-degenerate which means that we have an energy gap and this is one of the ways that the energy gap is going to come in and play a role and now we adiabatically evolve it within that lowest eigenstate which means that we're not including the possibility that the state could transition so if we evolve it it's got to stay in the same state of course add a different point in the parameter space and then up to the possibility of having a phase factor okay that's all it can do and we know how to treat adiabatic quantum evolution we can plug this into the Schrodinger equation so IH bar d by dt of this state has got to be the Hamiltonian with this time dependence on that state but if you just plug this in you can see that you have two ways that this d by dt can act on the state you're putting in it can bring down this phase it can be d theta by dt or it can act directly on that eigenstate because that is also a time dependent eigenstate and on the other side on the right hand side you're just getting the energy back out because you acted with the Hamiltonian now I multiply from the right by n by the same eigenstate and I use that these are normalised orthogonal eigenstates such that on this side multiplying by n is just giving me this H bar d theta by dt here I of course just have the en and I've taken this over to the other side now where I keep I change the sign because it's over on the other side the n d by dt n so I've just taken this simple idea plugged it into the Schrodinger equation and said let's see what happens well that's a differential equation that I can solve I can integrate that with respect to time and my phase this phase I don't know what it is has got to be the integral over time of the energy as a function dt prime so en rft prime dt prime and then this quantity here which is telling us about how the eigenstate is changing so what does this mean? well the first part you should all recognise is just the dynamical phase that's just the dynamical phase that we always get in quantum mechanics so when eigenstate evolves with e to the i e t over H bar the other part is something that you may or may not have seen in your undergraduate course and this part is what I'm most interested in this is going to be the very phase so you can see that we have this differentiation with respect to t prime so it's telling us about how those eigenstates are changing but to make it a bit clearer let's just change the variables to get out that time dependence so now instead we differentiate with respect to the parameter r and now outside we have a dr by dt prime okay so and then very roughly using physics mathematics then we see that what we end up with is a phase that depends on how the parameters change so it's only depending on the contour that we took in that parameter space not on the explicit time dependence so it doesn't matter if we went fast or we went slow it just matters where we went and this is why we call it a geometrical phase because it only depends on where we went in parameter space not on dynamically how fast we went okay so actually people knew this a long long time ago like this is really something that you can write down right from the beginning of quantum mechanics back in the 1920s but people thought ah well this doesn't really matter because in quantum mechanics we have gauge freedom we're free to choose the gauge of our eigenstates so if we change that gauge that means that we can multiply our eigenstates by a parameter dependent phase factor what happens to that very phase well this is an extra phase factor that we now have here that we put in to this and we can see we get the same state the same expression but now we also have this gradient of the gauge that we chose that gauge function and so if we just integrate that then you can see that if we choose these two bits of the gauge function correctly then surely we should always be able to cancel out that very phase and this be an important physical quantity because it's gauge dependent we just choose that chi at the end of our evolution and the chi at the beginning of the evolution so that we cancel this out and that's what they thought they just thought well this we can always make go away so it can't be physical and it took until Michael Berry in the 1980s to say wait a second what if I don't consider just any evolution consider evolution that goes in a circle that's a closed evolution where my starting and finishing point is the same thing well then I actually have a problem because I said that I was going to choose this gauge and I decided I wanted to choose a smooth gauge a gauge that led to a single definition of my eigenstates everywhere but now this condition the fact that I want this e to the i chi of r at t to be the same as e to the i chi of r at zero I want those two factors to be the same in order to have the same definition that to have my eigenstate single valued within this gauge that means that this difference between these two things has to be two to pi times an integer I've lost the ability to choose the gauge in order to get rid of that very phase and that means that the very phase is physical and gauge invariant, modulo two pi for a closed contour so it's not always gauge invariant we only really talk about very phases that are associated with these closed contours but that's already great and this is a new aspect of quantum mechanics that we'll see is very very important and actually of course is much wider than the narrow consequences and topological phases of matter that I'm going to be talking about today so I don't have that long left which is okay because I might have been going slowly but I'm doing about the pace I wanted to do so this is good so how do we think about this very phase well one of the things that I always like to picture in my head is parallel transport of a vector so oh yeah sorry that's zero oh sorry this is not the very so the very phase itself this is just the gauge function it's the gauge function condition so it's that the very phase is gauge invariant up to two pi yes no no no you have to do a closed contour and then you can get the gauge so once you do a closed contour you can have the very phase itself being equal to anything it's just that if you were to change the gauge you could get a two pi addition to that that would actually be the same physical thing which you can see as well because it's a phase factor yep oh yeah so I was just talking about this analogy with so this is how I like to visualise it so if you think about parallel transport of a vector that means that we take the surface and we move a vector according to particular mathematical rules on a closed contour and then we look when we come back to the same point whether that vector has rotated and actually one thing we know is that if we have a flat system a system with no Gaussian curvature then that vector will not rotate but if we have a system that has some curvature for instance a sphere then that vector does undergo a rotation and that rotation is related actually to the solid angle that is enclosed it's related to the Gaussian curvature of the surface and what we're doing here is kind of a bit like parallel transport we're actually doing a version of parallel transport but as is defined for eigen states and the associated fibre bundles so mathematically the maths is more involved but this is a good basic analogy if you want to I think in the next slide I have okay in a few slides then I will have a reference to Nakahara's geometry and topology and physics which is a great reference that talks about these ideas of the fibre bundle and how this is related but this is the picture that I have in mind and that inspires us to introduce some other concepts so just like that Gaussian curvature that led to a rotation of the vector we can talk about a berry curvature that leads to this berry phase okay so what am I actually doing here I'm just taking the expression for the berry phase and I'm saying let's call this part of the integrand something called a berry connection so you know the berry phase is this but I'm going to take this part of it and call it a berry connection which is telling us basically the parallel transport condition and that is related to a curvature so just as we could integrate the Gaussian curvature to get the rotation of the vector so we can think about integrating this berry curvature to get the berry phase and these are very nice subsidiary concepts to introduce as we shall see one of the important things to note is that this berry connection is actually gauge dependent so it's a really really tricksy thing you don't want to spend too much time talking about berry connections because they can be deceiving but the berry curvature is lovely because it is fully gauge invariant as you can see by just plugging in what a gauge transformation does into these quantities and one of the things that is very very powerful in particular in cold atoms is this idea that actually we can make a beautiful analogy here as well with magnetism so this berry phase is a phase that we get when we do this close contour and that is very similar to the phase that we would get the arrow off bone phase if we encircle a certain amount of magnetic flux so you can think about the berry phase as being a measure of how much flux of this magnetic flux is through this contour you can think about the berry connection as being like a magnetic vector potential magnetic vector potential is also gauge dependent so the analogy is very deep here as we'll see and you can think about this berry curvature as being like a magnetic field so these are some very very useful concepts and now we're going to see how we can push this one step further and that is to make the connection now back to topology so everything I just told you about was geometrical because you can see that what I'm talking about are local properties of this parameter space I'm not talking about a global topological invariant yet I'm talking about the local properties so what about global topology well in mathematics there's also a connection between local geometry and global topology which is the Gauss-Bonnet theorem and that is as stated here that the integral of the Gaussian curvature over a closed surface is related to the genus of that surface so this is the topological invariant of surfaces that we met earlier in this talk and actually we have an analogy here which is that the integral of the buried curvature over a closed surface is a topological invariant an integer called the first trend number okay so I'm technically at the end of my time so I can leave the derivation of exactly why this is and why this is an invariant to next time but yeah this subject then of where we'll pick up tomorrow will be how do we see that integrating the buried curvature over a closed surface has to give us an integer and then how do we see that that integer is the first trend number of those energy bands that I talked about at the beginning okay thank you very much