1 00:00:05,590 --> 00:00:14,590 Ladies and gentlemen, good afternoon. Welcome to the Physics Department and welcome to the 56 Churchill Simon Memorial Lecture. 2 00:00:15,550 --> 00:00:22,480 This lecture is endowed in the names of critics Alexander Lindemann, first discount of Churchill and France, 3 00:00:22,750 --> 00:00:30,700 and Simon Lindemann became the Dr. Leeds Professor of Experimental Philosophy in 1919 and head of the Clarendon Laboratory. 4 00:00:31,450 --> 00:00:42,370 Brunswick and Simon obtains his degrees in Berlin under Nernst and then was called by Lindemann in 1933 to the Clarendon Laboratory, 5 00:00:42,370 --> 00:00:51,880 where he became first reader, and then Dr. Police Professor and head of the Clarendon Laboratory himself in 1956, one month before his death. 6 00:00:53,590 --> 00:01:03,500 The first Churchill Simon Memorial Lecture was held in 1960, and this is a picture of the first 50 memorial lists, first with George Thomson. 7 00:01:04,150 --> 00:01:12,910 As you can see, much of the to of physics in the second part of 20th century is here with many Nobel Prize medallists, 8 00:01:12,910 --> 00:01:18,340 not only in physics, but also in medicine, crick and chemistry. 9 00:01:18,850 --> 00:01:28,810 Kroto. The 50th lecture was Michael Berry, and following him we had Michael Pepper in 2010, 10 00:01:29,500 --> 00:01:36,970 David GROSS 2011, Anthony Salinger 2012, and Robert Loughlin 2013. 11 00:01:37,540 --> 00:01:45,520 Last year we should have had Peter Higgs. Unfortunately, he cancelled too much of much to our regret, the very morning of the lecture. 12 00:01:47,020 --> 00:01:51,520 All these great names testify that physics is full of discoveries. 13 00:01:52,810 --> 00:02:00,880 However, very few such discoveries qualify as paradigm shifts to use the language introduced by Thomas KUHN. 14 00:02:01,960 --> 00:02:06,430 Discoveries such as the laws of electromagnetism. Special relativity. 15 00:02:07,120 --> 00:02:14,050 Quantum mechanics are not only groundbreaking, but completely change the way we view our world as physicists. 16 00:02:15,580 --> 00:02:22,360 One such paradigm shift in the 20th century, perhaps most clearly associated with Soviet physicists left Londo, 17 00:02:22,750 --> 00:02:29,380 was the realisation that changes in symmetry are ubiquitous in describing and producing physical phenomena 18 00:02:29,650 --> 00:02:37,240 as simple as the crystallisation of water in twice and as complex as superconductivity or the Higgs boson. 19 00:02:38,320 --> 00:02:45,940 Another paradigm shift in the 21st century, perhaps, unbeknown to most of the general public, has occurred, 20 00:02:47,380 --> 00:02:56,410 whereby a new concept called topology is taking a role that is not too dissimilar from that taken by symmetry previously. 21 00:02:57,340 --> 00:03:01,240 This paradigm shift is most clearly associated with today's lecture. 22 00:03:02,110 --> 00:03:10,540 It is therefore my great pleasure and honour to introduce to you the 56th Chair, Will Solomon Memorial Lecture. 23 00:03:10,840 --> 00:03:21,460 Professor Charles Keene. Professor Keene is the Class of 1956 endowed term chair, professor of Physics at the University of Pennsylvania. 24 00:03:21,550 --> 00:03:23,350 And I had to read that it was quite complicated. 25 00:03:24,370 --> 00:03:34,270 He completed his bachelor degree in physics at the University of Chicago in 1985 and at the MIT in 1989. 26 00:03:34,630 --> 00:03:40,480 Prior to joining the faculty at Pennsylvania, he was a postdoc at IBM. 27 00:03:40,480 --> 00:03:48,190 T.J. Watson Research Centre. Professor Cahn obtained many awards for his work. 28 00:03:48,820 --> 00:03:53,320 He was the director prize winner in 2012. 29 00:03:53,680 --> 00:04:00,760 Two that together with the shoeshine Jane and Dunkel Haldane for their groundbreaking work on 2D and 3D topological insulators. 30 00:04:02,350 --> 00:04:08,470 He was chosen in the same year for the inaugural class in mathematics and physics for the same assignment investigators. 31 00:04:08,890 --> 00:04:16,990 And he shared the following year the Physics Frontier prizes with Lawrence Malcolm intrusion for their work on topological insulators. 32 00:04:17,710 --> 00:04:23,260 He will speak to us today about topological boundary modes for quantum electronics to classical mechanics. 33 00:04:24,340 --> 00:04:39,300 So please welcome Professor Charles Cahn. Well, thank you very much. 34 00:04:39,320 --> 00:04:42,290 It's really quite an honour to be here. 35 00:04:42,320 --> 00:04:50,990 And and it's been a real pleasure spending the week here visiting both some old friends of mine, as well as making some new friends, 36 00:04:50,990 --> 00:04:59,809 and also getting the opportunity to experience firsthand some of the wonderful traditions that you have at this historic institution. 37 00:04:59,810 --> 00:05:06,680 So it's really been a pleasure to to come here. And so I'd like to thank Paolo for for introducing my subject, 38 00:05:06,680 --> 00:05:15,110 because the ideas of symmetry and topology are indeed two of the sort of building blocks, 39 00:05:15,110 --> 00:05:20,030 the foundational tools that we have for understanding matter. 40 00:05:20,240 --> 00:05:29,570 Now symmetry, of course, has to do with what you can do to a system that doesn't change it, so you can rotated or translated. 41 00:05:29,750 --> 00:05:33,590 And symmetry has been one of the guiding principles in physics. 42 00:05:33,590 --> 00:05:37,040 When you're a student in physics, it's the first thing that you learn that you know, 43 00:05:37,070 --> 00:05:40,879 if you can understand the symmetries of problem, you can simplify it and you can understand it. 44 00:05:40,880 --> 00:05:51,690 And so, so and and and as Landau pointed out, symmetry is a very powerful tool for distinguishing the fundamental phases that matter can exist in. 45 00:05:52,110 --> 00:05:57,470 Okay, so. So topology is an equally deep and equally powerful idea. 46 00:05:58,250 --> 00:06:02,960 And, and rather than thinking about what, what you can do to a system that keeps things the same, 47 00:06:03,170 --> 00:06:08,180 it addresses the question what stays the same when you change a system? 48 00:06:08,930 --> 00:06:14,180 And so so for instance, if you think about a the surface of an orange is kind of like a sphere. 49 00:06:14,420 --> 00:06:20,659 Well, you can squeeze the split step on the orange and squeeze it down and make it look like the surface of a bowl with, 50 00:06:20,660 --> 00:06:23,270 you know, and you could do that continuously and nothing changes. 51 00:06:24,170 --> 00:06:30,230 So there's a sense in which the sphere, the, you know, the surface of surface of the orange and the surface of the ball are the same. 52 00:06:30,380 --> 00:06:36,800 OC Even though there's no symmetry, you know, between them, but they're different from a doughnut. 53 00:06:37,190 --> 00:06:41,690 To get to a doughnut, you'd have to poke a hole in the atmosphere, and that's against the rules. 54 00:06:41,870 --> 00:06:46,940 Okay, but a doughnut is kind of like the surface of a coffee cup, okay? 55 00:06:47,030 --> 00:06:51,800 Because you can stretch it and mould it, and the whole of the doughnut turns into the handle of the coffee cup. 56 00:06:51,840 --> 00:06:58,610 Okay. And so this idea of topology asking what what are the sort of fundamental things that stay the same when you change 57 00:06:58,610 --> 00:07:07,399 something is an is is a an additional powerful tool that we have for now understanding the phases of matter. 58 00:07:07,400 --> 00:07:15,729 And so this is a subject which over the past ten or 15 years has really developed a lot. 59 00:07:15,730 --> 00:07:19,700 And really, it's it's it's our, you know, so of course, symmetry and topology are old ideas. 60 00:07:19,700 --> 00:07:24,200 They've they've been present in physics for for for many, many years. 61 00:07:24,350 --> 00:07:32,030 What's new is the is our appreciation of the interplay between symmetry and topology and in particular for understanding the quantum phases of matter. 62 00:07:32,030 --> 00:07:41,240 And so what I'd like to do today is to describe to you some various situations where where this is crucially important. 63 00:07:41,420 --> 00:07:47,660 And in particular, I'd like to describe the sort of new quantum electronic phases of matter that we have discovered. 64 00:07:48,350 --> 00:07:56,720 And, and explain to you how symmetry and topology sort of intertwine to to to give us an understanding of those materials. 65 00:07:57,140 --> 00:08:00,680 And after I do that, though, this is going to be a little bit of a schizophrenic talk, 66 00:08:00,680 --> 00:08:05,180 because I'm going to shift gears and talk about a completely different set of physics problems, 67 00:08:05,600 --> 00:08:10,639 not not quantum classical physics problems, where remarkably, 68 00:08:10,640 --> 00:08:15,560 the same kind of mathematical ideas that we use to understand these materials show 69 00:08:15,560 --> 00:08:21,500 up in these classical mechanical modes or what I will call ISO static lattices. 70 00:08:21,620 --> 00:08:28,729 Okay. And so, so I'm going to show to you show you the sense in which these two sort of look the same and and argue to you that that that really 71 00:08:28,730 --> 00:08:35,540 they can be understood with similar using the similar sort of basic principles of physics that that of symmetry and topology. 72 00:08:36,200 --> 00:08:46,150 So, so my contributions in this in this field sort of started off about ten years ago in a collaboration I had with Jean Mihaly, 73 00:08:46,160 --> 00:08:53,330 who's my colleague at the University of Pennsylvania. And then over the over the years, I've been joined by a number of students and postdocs. 74 00:08:54,200 --> 00:09:01,450 The most recent work was a collaboration with another colleague of mine at the University of Pennsylvania, Tom Levinsky. 75 00:09:01,490 --> 00:09:09,830 So this last part of the talk is really dedicated to time. Okay, so let me start off with talking about electronic quantum electronic phases. 76 00:09:09,830 --> 00:09:14,270 And and I want to start with the most basic phase of matter. 77 00:09:14,360 --> 00:09:19,100 Okay. And and the most basic phase of matter is the the insulating state. 78 00:09:19,280 --> 00:09:24,679 Okay. Now, you know, you might think the insulating state is the most boring state. 79 00:09:24,680 --> 00:09:28,250 It's basically the state where the electrons do nothing. 80 00:09:28,550 --> 00:09:32,240 Okay. But it's something that we tell our students about. 81 00:09:32,240 --> 00:09:36,500 You know, in in beginning physics, what we say is that in an insulator. 82 00:09:38,200 --> 00:09:43,240 It cannot conduct electricity because the electrons are bound to the atoms and they're stuck to the atoms. 83 00:09:43,660 --> 00:09:46,840 And so. So every electron is sort of stuck. It can't move. 84 00:09:47,110 --> 00:09:56,350 Electricity cannot flow. Now, actually, this fact that the electrons are stuck to the atoms is a uniquely quantum mechanical fact. 85 00:09:56,480 --> 00:10:05,110 Okay. And so, so, so. And the reason for that is that in quantum mechanics, due to the motion of electrons in around an atom, 86 00:10:05,260 --> 00:10:11,140 quantum mechanics tells you that the energies that electrons can have are quantised. 87 00:10:11,980 --> 00:10:19,240 They can only have specific energies. And so what happens in an atom, in a in an inert atom, say, like like argon, 88 00:10:19,360 --> 00:10:28,509 is that you have all of the electrons occupy one set of one set of states, and they're separated by a large energy gap to the next state. 89 00:10:28,510 --> 00:10:30,200 You can put an electron it. Okay. 90 00:10:30,370 --> 00:10:37,810 And so if you have a situation like this, then the electrons have a huge energy barrier to get over in order for them to do anything. 91 00:10:38,170 --> 00:10:40,030 Okay. And so for that reason, 92 00:10:40,660 --> 00:10:47,980 a crystal made out of these argon atoms would be electrically inert because it takes a huge energy in order to to dislodge an electron. 93 00:10:48,280 --> 00:10:54,129 Now, of course, a crystal of, you know, inert argon atoms is not a very interesting material, 94 00:10:54,130 --> 00:11:00,850 but things get more interesting if you if you bring the atoms closer together and you allow them to start chemically bonding with one another. 95 00:11:01,030 --> 00:11:07,960 And in that case, one has a more interesting situation where the different atoms interact with one another. 96 00:11:08,350 --> 00:11:19,360 And so so the the theory that describes this was really one of the crowning triumphs of quantum mechanics in the 20th century. 97 00:11:19,720 --> 00:11:29,200 And so this theory for understanding the electronic structure of crystalline materials is called the band theory of solids. 98 00:11:29,350 --> 00:11:35,079 Okay. And and this is really probably one of the most consequential theories that of matter that 99 00:11:35,080 --> 00:11:40,239 we have is the theory that's responsible for our understanding of semiconductor materials, 100 00:11:40,240 --> 00:11:44,540 for example. It's the theory that underlies this laser pointer. 101 00:11:44,540 --> 00:11:55,389 It underlies the cell phone in your pocket. All of the information technology that that that that we are experiencing today ultimately results from 102 00:11:55,390 --> 00:12:01,030 this from the deep understanding of of materials that we have based on the band theory of solids. 103 00:12:01,240 --> 00:12:09,880 And so so what the band theory tells you is that, you know, you start off and you have energy levels that are separated by an energy gap. 104 00:12:10,000 --> 00:12:14,290 And what happens when you make them in a crystal is they broaden out and form a band of energies. 105 00:12:15,550 --> 00:12:24,910 And in an insulator, the the occupied energy states are separated by an energy gap from the empty energy state. 106 00:12:24,920 --> 00:12:30,340 So an energy gap separates the so-called conduction band from the valence band. 107 00:12:31,090 --> 00:12:37,899 And so so due to the existence of this energy gap, this insulator is electrically inert. 108 00:12:37,900 --> 00:12:47,830 For the same reason this was OC and the band theory of Solids gives us a very detailed understanding of the of the structure of the states in this, 109 00:12:47,830 --> 00:12:52,930 in this valence band that's enabled enables, you know, lots of technological advances. 110 00:12:53,320 --> 00:13:03,520 Now, one point that I want to make, though, is that there's a sense in which this is kind of the same as this in the following, 111 00:13:03,520 --> 00:13:07,030 in the sense of topology that I introduced before, 112 00:13:07,150 --> 00:13:14,470 which you can imagine sort of smoothly deforming this one into this one, making the atoms get a little bit further apart, further apart. 113 00:13:14,650 --> 00:13:22,390 And, and, and as you smoothly deform from this to this, the energy gap would get bigger maybe, but nothing drastic would happen along the way. 114 00:13:22,690 --> 00:13:27,580 Okay. So there's a sense in which the the insulating state of a say, 115 00:13:27,760 --> 00:13:35,740 intrinsic semiconductor is kind of the same as a the insulating state of of of a, you know, gas of inert atoms. 116 00:13:36,010 --> 00:13:39,010 Actually, it's also kind of the same as the vacuum. 117 00:13:39,610 --> 00:13:47,890 Okay. Now, you know, the vacuum actually is not as empty as you might first think. 118 00:13:48,850 --> 00:13:54,580 So if you think in terms of the relativistic theory of quantum mechanics, Dirac's theory of quantum mechanics, 119 00:13:55,450 --> 00:14:05,920 the vacuum has a valence band to okay the valence band for the for the vacuum is if it is the positron. 120 00:14:05,930 --> 00:14:11,860 So if I remove an electron from this valence band that's what the antiparticle of the electron is the positron. 121 00:14:12,610 --> 00:14:16,990 And so the only difference is that the energy gap here is a million times bigger. 122 00:14:17,770 --> 00:14:20,770 Okay? So there's a sense in which all three of these are the same. 123 00:14:21,280 --> 00:14:32,860 Okay? And, and that sameness is where the idea of topology enters the discussion of the band theory of solids. 124 00:14:32,950 --> 00:14:43,100 Okay. And what's remarkable is. That these ideas of topology and band theory were left undiscovered for so long. 125 00:14:43,600 --> 00:14:46,970 It was a treasure that was sort of waiting to be uncovered. So. 126 00:14:47,760 --> 00:14:50,890 So. So the question you can ask. 127 00:14:51,640 --> 00:15:01,990 So. So. I will say that insulators are topologically equivalent if they can be smoothly deformed into one another without closing the energy gap. 128 00:15:02,650 --> 00:15:05,740 Okay. So that poses the question. 129 00:15:06,340 --> 00:15:16,330 Are there any other kinds of topological phases which have an energy gap that you can't get to from a single from a trivial insulator? 130 00:15:16,450 --> 00:15:21,070 Is there an analogue of the doughnut? Okay. And the remarkable answer is that, yes, there is. 131 00:15:21,220 --> 00:15:24,760 And these electronic phases are something that are very interesting to study. 132 00:15:25,330 --> 00:15:32,569 So so what I want to do is sort of describe to you the the simplest version of this in my approach to these kinds of things, 133 00:15:32,570 --> 00:15:36,280 that you always try to find the simplest example of something first. 134 00:15:36,280 --> 00:15:40,930 And the simplest example of this actually has a long history. 135 00:15:41,830 --> 00:15:47,229 The so the simplest example is what you can have if you have a, a system which is one dimensional. 136 00:15:47,230 --> 00:15:52,030 So it's defined on a line. And so the system I have in mind is a conducting polymer. 137 00:15:52,030 --> 00:15:58,180 So this is a chain of atoms, it's a chain of card carbon and hydrogen atoms called poly acetylene. 138 00:15:58,690 --> 00:16:02,049 And poly suddenly is an important material. 139 00:16:02,050 --> 00:16:06,010 It's sort of the first the simplest example of a conducting polymer now. 140 00:16:06,160 --> 00:16:12,140 So so if you just have a chain of these carbon atoms, each one has has an electron on it. 141 00:16:12,340 --> 00:16:24,010 And and so what happens is that if if it's perfectly periodic, then what you have is you have a an energy band that is half filled. 142 00:16:24,870 --> 00:16:32,470 And so this means that there is no energy gap separating the occupied electronic states from the empty electronic state. 143 00:16:32,480 --> 00:16:36,880 So electrons can sort of move around with little cost and energy. 144 00:16:36,890 --> 00:16:44,530 So the thing. So it is a conductor. Okay. Now, the thing that's interesting about this, though, is that this is not what poly acetylene wants to do. 145 00:16:45,040 --> 00:16:49,690 What Pauli acetylene would rather do is it would rather die, 146 00:16:49,760 --> 00:16:57,700 arise and have the pairs of atoms move a little bit closer to each other and and a little bit further apart. 147 00:16:57,700 --> 00:17:01,810 So, so the so it forms dimers where the bonds are a little bit stronger. 148 00:17:02,620 --> 00:17:11,290 And by doing that, what it does is it opens up an energy gap so that all of the occupied states go down in energy. 149 00:17:11,560 --> 00:17:18,070 Okay. And so so that's the reason it likes to do that because because the system gains energy by by opening this energy gap, 150 00:17:18,070 --> 00:17:22,600 having all the occupied states go down. But the interesting thing is that now it's an insulator. 151 00:17:22,870 --> 00:17:28,090 Now there's an energy gap which which separates the occupied states from the empty states. 152 00:17:28,480 --> 00:17:36,160 Okay. So but but that's the thing that's nontrivial about this is that there are two ways you can do it. 153 00:17:37,000 --> 00:17:41,590 You can either demonise going short, long, short, long. 154 00:17:41,650 --> 00:17:44,930 Okay. And that would be if I move the green atoms to the left. 155 00:17:45,400 --> 00:17:50,440 Okay. Or I could die. My eyes going long, short, long, short. 156 00:17:50,920 --> 00:17:55,570 Moving the green atoms to the right. Okay, those are both insulators. 157 00:17:56,110 --> 00:18:00,340 But what I want to argue to you is that in this sense that I defined, 158 00:18:00,490 --> 00:18:09,940 they are topologically distinct insulators in the following sense that that if I tried to smoothly go from this situation to this situation, 159 00:18:10,270 --> 00:18:13,520 I'd have to go through here where the energy gap is zero. Okay. 160 00:18:13,750 --> 00:18:17,740 And so the A phase and the B phase are separated. 161 00:18:18,640 --> 00:18:21,490 If I try to go if I try to tune myself from one to the other, 162 00:18:21,760 --> 00:18:28,300 I have to go through a quantum phase transition, a critical point where the energy gap goes to zero. 163 00:18:29,230 --> 00:18:34,510 Okay. I mean, because and so that's exactly what I mean by topological. 164 00:18:34,510 --> 00:18:38,140 Like, you know, all the Fe all the states over here are topologically equivalent. 165 00:18:38,380 --> 00:18:42,070 All the states over here are topologically equivalent. But these and these are different. 166 00:18:42,370 --> 00:18:45,759 Okay. And so so something one can do. 167 00:18:45,760 --> 00:18:49,209 And I don't want to sort of delve into the mathematics of how one describes this, 168 00:18:49,210 --> 00:18:58,900 but by understanding the structure of the electronic states in the valence band, one can define what's called a topological invariant. 169 00:18:58,990 --> 00:19:06,370 So this is something that you can compute. And the way you can think about this topological invariant is that it is a kind of winding number. 170 00:19:06,380 --> 00:19:11,220 So so if you you can imagine if you have a rubber band wrapped around a pole, okay, 171 00:19:11,470 --> 00:19:15,130 you could distinguish this from you can distinguish whether it's wrapped around or not. 172 00:19:15,410 --> 00:19:18,700 In fact, it could wrap around any integer number of times. 173 00:19:18,880 --> 00:19:25,540 And the distinction between the A and the B phase really is sort of like this this this winding number. 174 00:19:25,790 --> 00:19:30,300 Okay. I will say that this this there's there's a there's a cow. 175 00:19:30,310 --> 00:19:34,720 I always have a little star here. So there's always a little bit of a caveat, a technical caveat with this. 176 00:19:34,930 --> 00:19:42,419 There's an assumption that I made a. Out this, which is that that this polio suddenly it has a has a actually an extra symmetry. 177 00:19:42,420 --> 00:19:49,230 So symmetry is actually playing a role here. And that symmetry is actually an important one which is going to show up again, 178 00:19:49,410 --> 00:19:53,420 which is a symmetry that relates to the conduction band of the Valence band. 179 00:19:53,430 --> 00:19:59,850 So in the simplest picture of polys settling the conduction band and the Valence band are mirror images of each other. 180 00:19:59,850 --> 00:20:03,840 So everything going on at a positive energy is also happening at a negative energy. 181 00:20:04,290 --> 00:20:08,850 So, so battery case. So, so the two phases are topologically the same. 182 00:20:08,860 --> 00:20:12,300 So you could ask yourself, okay, what, what consequence does that have? 183 00:20:13,050 --> 00:20:18,510 And the most important consequence of this topological distinction is what happens 184 00:20:18,720 --> 00:20:23,700 on the boundary where the two distinct topological phases meet each other. 185 00:20:23,940 --> 00:20:28,140 Okay? And so you can imagine, let's suppose you have a phase on the left in the B phase on the right. 186 00:20:28,380 --> 00:20:35,820 Well, a simple way of thinking about this is just think about the limit, where these bonds are infinitely strong and these bonds are infinitely weak. 187 00:20:36,030 --> 00:20:40,440 Then you have an atom that's left over in the middle here, and it's not talking to anybody. 188 00:20:41,370 --> 00:20:45,809 And so you have a big energy gap from from from from on both sides. 189 00:20:45,810 --> 00:20:50,640 But then there's this one state that's left over in the middle here that's at zero. 190 00:20:50,850 --> 00:20:54,690 Okay. Now you can ask yourself, okay, what happens if I turn these bonds up? 191 00:20:55,530 --> 00:21:02,940 Okay. But I told you that we have this symmetry, that whatever is happening in a positive energy is also happening at a never negative energy. 192 00:21:03,180 --> 00:21:06,480 And that means that this state of zero can't go anywhere. 193 00:21:06,810 --> 00:21:14,160 Okay, because if it tried to move up, then then it would break this, this particle, hold this, this symmetry under egos to mine to see. 194 00:21:14,250 --> 00:21:22,530 Okay, so, so this this zero mode here is really a topological zero mode in the sense that you can't get rid of it. 195 00:21:23,670 --> 00:21:33,420 You inevitably have it. If you have an interface between this A and B phase, now, this is a fact which has a long history in physics, 196 00:21:33,420 --> 00:21:38,070 and it's something which has been discovered and rediscovered many, many times over the years. 197 00:21:38,340 --> 00:21:44,850 So I think the first discovery of this idea of the topological boundary mode was by the field theorists, 198 00:21:44,850 --> 00:21:51,030 in particular Roman Jacques-yves, and he was interested in certain sort of one dimensional field theory problems. 199 00:21:51,660 --> 00:22:03,590 And then some time later, Bob SCHIEFFER and his collaborators were interested in poly acetylene and were interested in particular these these domains, 200 00:22:03,600 --> 00:22:06,690 these domain wall states in polish acetylene. 201 00:22:06,690 --> 00:22:10,410 And they sort of rediscovered this this idea that you have this topological zero map. 202 00:22:10,890 --> 00:22:19,190 Now, in the years since, what we have learned is that there are many, many more examples of this phenomena. 203 00:22:19,200 --> 00:22:26,010 And so, so so I want to sort of broaden this idea to what I'll call the the bulk boundary correspondence, which says in general, 204 00:22:26,160 --> 00:22:37,260 at the boundary between two topologically distinct insulating phases, there exist some topologically protected low energy states. 205 00:22:37,710 --> 00:22:42,030 Okay? And so these top these, these, these low energy states are states that you can't get rid of. 206 00:22:43,150 --> 00:22:46,360 Okay. So. So there are many, many examples of this. 207 00:22:48,070 --> 00:22:53,350 Probably the most famous one is what is called the Quantum Hall effect. 208 00:22:54,100 --> 00:22:56,900 And so the Quantum Hall effect is something again, it happens to electrons. 209 00:22:57,190 --> 00:23:03,370 It happens to electrons when you combined confine them to a two dimensional plane. 210 00:23:03,380 --> 00:23:11,140 So you have electrons that can live on an interface, a two dimensional interface between two materials, and you put them in a strong magnetic field. 211 00:23:11,170 --> 00:23:16,240 Now, what a magnetic field does to electrons is it makes them want to go around in circles. 212 00:23:16,630 --> 00:23:19,600 So that's what electrons do in a magnetic field. 213 00:23:19,750 --> 00:23:25,480 And, you know, this motion going around in circles, it's kind of like the same kind of motion that electrons do when they go around in an atom. 214 00:23:25,930 --> 00:23:34,719 Right. And so maybe it's not a surprise that when you think about the effect of quantum mechanics on this circular motion of the electrons, 215 00:23:34,720 --> 00:23:39,010 it does kind of the same thing that it did in an atom. It makes the energy levels quantised. 216 00:23:39,970 --> 00:23:45,040 And so these quantised energy levels in a magnetic field are called landau levels. 217 00:23:45,310 --> 00:23:54,040 And you can have a situation where you have some number of landau levels are occupied and then the next the higher landau levels are empty. 218 00:23:54,890 --> 00:23:57,570 And so so this is kind of like an insulator, right? 219 00:23:57,580 --> 00:24:05,290 You have an energy gap that separates the occupied electronic states from the empty electronic states. 220 00:24:05,430 --> 00:24:09,760 Okay, so it's just like an insulator, but it's not an insulator. 221 00:24:10,040 --> 00:24:17,470 Okay. It turns out if you apply an electric field, if you try to put a put a voltage on on on this state, 222 00:24:17,650 --> 00:24:20,950 then what the electrons who are going around in a circle, if you apply an electric field, 223 00:24:20,950 --> 00:24:28,900 the electrons start moving perpendicular to the electric field. That's what happens if you have motion in a cross E and B field. 224 00:24:29,770 --> 00:24:35,770 And so so actually this defines a. 225 00:24:36,280 --> 00:24:39,310 So you say you put a field this way. You've got a current this way. 226 00:24:39,430 --> 00:24:43,720 This defines what's called the hall effect. You get a current perpendicular to the field. 227 00:24:43,870 --> 00:24:51,730 And in fact, what's remarkable is if you measure this current, if you measure the hall current, the current going perpendicular, 228 00:24:51,880 --> 00:25:05,920 it defines a conductivity, if you will, which is incredibly accurately quantised in units of this fundamental unit of conductance. 229 00:25:06,220 --> 00:25:10,900 So this is a combination of fundamental constants, you know, the electric charge and flex constant. 230 00:25:11,140 --> 00:25:15,760 This defines a unit for the resistance or the or the conductance. 231 00:25:16,000 --> 00:25:24,220 And what is measured is that you measure these incredibly sharp plateaus in the hall conductance, 232 00:25:24,430 --> 00:25:30,400 where this integer here is something that you can measure to one part in a billion. 233 00:25:31,300 --> 00:25:36,820 It's an incredibly accurate quantisation. And and so so what's the origin of this? 234 00:25:36,970 --> 00:25:44,260 Well, the origin of it is really topology. What you are measuring here is actually a topological invariant. 235 00:25:44,890 --> 00:25:54,160 And so this this idea of a topological invariant in the Quantum Hall effect was really first discovered by one of my one of my heroes in physics, 236 00:25:54,340 --> 00:25:57,999 David Thalis and his collaborators. 237 00:25:58,000 --> 00:26:07,000 And he pointed out that the hall conductivity is, in fact a topological invariant that has a number in the mathematics, 238 00:26:08,590 --> 00:26:14,800 in mathematics called the chern number that characterises a two dimensional insulator, basically. 239 00:26:14,950 --> 00:26:18,760 Okay. And so, so, so. So you get this. So, so, so. 240 00:26:18,910 --> 00:26:22,150 So this quantised hall conductivity really is a topological invariant. 241 00:26:22,330 --> 00:26:32,320 Now, as I said, this topological class classification has consequences on the boundary where you have one topological classmates another. 242 00:26:32,440 --> 00:26:37,510 So if you have an integer, a quantum hall state here and the vacuum which is an equal zero, 243 00:26:37,780 --> 00:26:42,909 then what happens on the on the edge on the boundary is actually a funny, familiar phenomenon. 244 00:26:42,910 --> 00:26:46,590 In the Quantum Hall effect, you get a what's called an edge state. 245 00:26:46,600 --> 00:26:50,560 And so if you think about the motion of the electrons, the electrons are going around in circles in the magnetic field. 246 00:26:50,800 --> 00:26:56,110 Well, if you put a wall here, then the electrons can bounce off the wall and then they can skip along. 247 00:26:56,620 --> 00:27:05,470 Okay. So they undergo a motion, you know. But the thing that's kind of neat about this motion is it only goes one way, okay? 248 00:27:05,590 --> 00:27:12,580 It only goes from left to right. So it's sort of like it's sort of like a one way street for the electrons. 249 00:27:13,000 --> 00:27:21,660 Okay. So if you solve this problem, you know, with quantum theory, then then you still you then you discover these, these one way electrons. 250 00:27:21,670 --> 00:27:31,600 And in quantum theory, we give them a fancy or name, we call them one dimensional chiral, Dirac, fermions, and and these one dimension, 251 00:27:31,780 --> 00:27:39,190 these one way electronic modes are remarkable because what you know is if you're in one of these modes and you're an electron, 252 00:27:39,490 --> 00:27:49,570 you have no choice but to go forward. And so that means that if an electron comes in over here, it comes out over here with 100% probability. 253 00:27:50,350 --> 00:27:54,820 And this perfect transmission of these chiral edge states is really underlies 254 00:27:55,030 --> 00:27:58,960 the fact that you can measure this hall conductivity to one part in a billion. 255 00:27:59,470 --> 00:28:05,380 So there's there's something very deep about the topological protection of these chiral edge states. 256 00:28:06,410 --> 00:28:17,960 Okay. So now for many years it was believed that this was kind of the only non-trivial topological states that you could have. 257 00:28:18,510 --> 00:28:24,500 And it's a little bit of a contrived situation because because, you know, these edge states, they go one way. 258 00:28:25,070 --> 00:28:32,960 If you were to take a motion picture of this and run it backwards, then the edge states would be going the other direction. 259 00:28:33,420 --> 00:28:35,960 Right. If you ran it, ran the motion picture backwards. 260 00:28:36,320 --> 00:28:49,640 And so what this means is that this this this this motion of the electrons violates the symmetry under the reversal of time. 261 00:28:50,180 --> 00:28:56,480 So time reversal symmetry basically ask you whether whether if you run the motion picture backwards, whether things look the same or not. 262 00:28:56,900 --> 00:29:02,350 Now, the laws of physics. Are invariant under time reversal symmetry. 263 00:29:02,350 --> 00:29:09,460 If you if you had ran your motion picture backwards, what you saw would obey the laws of physics. 264 00:29:10,090 --> 00:29:11,680 Okay. But here, 265 00:29:11,680 --> 00:29:18,129 what the reason time reversal symmetry is violated is because you have to apply a magnetic field and a magnetic field violates time reversal, 266 00:29:18,130 --> 00:29:23,770 symmetry. And so. So the question one could ask is, what if you don't have a magnetic field? 267 00:29:23,920 --> 00:29:27,640 What if you do have symmetry under the reversal of time? 268 00:29:27,760 --> 00:29:34,630 Could there be anything interesting left? Okay, so this chern number dn would have to be equal to zero. 269 00:29:35,110 --> 00:29:37,900 Okay. And so for many years it was believed there was nothing left. 270 00:29:38,080 --> 00:29:43,240 But in the last ten years we've discovered that in fact there is something interesting that is left. 271 00:29:43,840 --> 00:29:50,320 So, in fact, if you have time reversal symmetry, the symmetry under running the motion picture backwards, 272 00:29:50,500 --> 00:29:56,410 then in fact there are actually two distinct classes of insulators. 273 00:29:56,920 --> 00:30:03,400 There is a trivial insulator, which is like the vacuum, but there is also a topological insulator. 274 00:30:04,000 --> 00:30:06,010 And unlike the Quantum Hall State, where, 275 00:30:06,010 --> 00:30:14,500 where there's sort of you have an equals 1 to 3 for any integer here there's just zero or one is az2 topological invariant. 276 00:30:15,190 --> 00:30:23,590 And and the the interesting thing is what happens on the boundary between the trivial and the nontrivial and on the boundary what, 277 00:30:23,650 --> 00:30:29,979 what has conducting edge states which are sort of like two copies of the edge states that you have in the Quantum Hall. 278 00:30:29,980 --> 00:30:32,860 Fact, you have one that's going one way and one that's going the other way. 279 00:30:32,860 --> 00:30:38,290 So so if I, if I, if I run it backwards in time, then, you know, it still looks the same. 280 00:30:38,980 --> 00:30:49,750 Okay. Now, so, so this is a phase of matter that we it's actually so this is a subject which proceeded in a rather unusual fashion. 281 00:30:49,900 --> 00:30:58,210 We actually realised theoretically that this state of matter could exist before we had seen it in the real world. 282 00:30:59,050 --> 00:31:04,420 And so there are various, you know, so once we knew it was possible, then the question is how can you make it? 283 00:31:04,660 --> 00:31:13,540 And, and so, so the way that one can make it is by making quantum well structures out of these materials Mercury, Telluride and cadmium, Telluride. 284 00:31:13,540 --> 00:31:16,270 These are these are sort of well known semiconductor materials. 285 00:31:16,480 --> 00:31:24,100 And and to cut a long story short, shortly after the prediction that that this phase was possible, 286 00:31:25,240 --> 00:31:35,320 experiments were done on these structures which which confirmed that that this structure did, in fact, have these these special kinds of edge states. 287 00:31:35,590 --> 00:31:39,550 Okay. So this is what happens in a two dimensional insulator. 288 00:31:39,910 --> 00:31:43,030 There's also a three dimensional topological insulator. 289 00:31:43,570 --> 00:31:50,560 So in three dimensions, again, there's there are two classes of insulators and in the non-trivial insulator, 290 00:31:50,650 --> 00:31:58,450 again, it's an insulator on the interior, but on the surface it is a very special kind of conductor. 291 00:31:59,230 --> 00:32:04,210 Okay. And it's and it's a special kind of conductor where the conducting state, you can't get rid of it. 292 00:32:04,600 --> 00:32:04,940 Okay. 293 00:32:05,080 --> 00:32:15,760 So so so in a sense, these the the, the, the, the electrical conduction on, on the surface of this topological insulator is topologically protected. 294 00:32:15,850 --> 00:32:21,970 Okay. And again, the, the the theoretical predictions of these of this state came first. 295 00:32:22,210 --> 00:32:26,290 And then after the predictions and the predictions of real materials came experiments. 296 00:32:26,290 --> 00:32:31,389 In this case, this is an experiment called angle resolved photo emission spectroscopy, 297 00:32:31,390 --> 00:32:36,160 which is a way of measuring the electronic structure of the surface of a material. 298 00:32:36,310 --> 00:32:43,150 And what one observe here observed here is sort of almost a textbook perfect version of what's called a surface Dirac cone, 299 00:32:43,150 --> 00:32:46,390 which is the prediction for what the electronic structure of the surface is. 300 00:32:46,930 --> 00:32:54,790 And so so the discovery of these materials has sort of opened a floodgate of, of, of activity in both experimental and theoretical physics. 301 00:32:54,790 --> 00:32:59,710 And so this is, you know, so these materials are very easy to to study. 302 00:32:59,920 --> 00:33:02,770 And there's there's really a lot going on. So so what? 303 00:33:03,010 --> 00:33:13,360 So I want to give one more example of a topological boundary mode that can occur in quantum electronic systems. 304 00:33:13,780 --> 00:33:21,610 And, and so this is one final example, which is what is called topological superconductivity now. 305 00:33:21,700 --> 00:33:25,050 Now superconductivity. This is a rather advanced topic in physics. 306 00:33:25,070 --> 00:33:31,959 I'm not going to I'm not going to be able to explain to you in detail what superconductivity is. 307 00:33:31,960 --> 00:33:42,550 But but but let me just say that a superconductor, if one uses it, thinks of in terms of what's called the BCS model of a superconductor. 308 00:33:42,670 --> 00:33:45,420 A superconductor is really just like an insulator. Okay? 309 00:33:45,760 --> 00:33:54,130 It has a conduction band and a valence band that are separated by an energy gap in just the same way as as as in an insulator. 310 00:33:54,340 --> 00:33:58,840 Okay. But the price that you pay for having this simple picture. 311 00:33:59,700 --> 00:34:08,340 Is that is that the conduction band and the Valence band actually really are mirror images of each other. 312 00:34:08,380 --> 00:34:12,690 Okay. You remember when I introduced this idea of the mirror, the particle hole symmetry before? 313 00:34:12,810 --> 00:34:19,860 It was sort of an approximate symmetry here. It's an intrinsic symmetry which relates the conduction band to the valence band. 314 00:34:19,980 --> 00:34:29,370 And in fact, even more the Conduction Band of the Valence Band are redundant in the following sense that if I put an electron in the, 315 00:34:29,370 --> 00:34:38,580 you know, I can add electrons to the conduction band. Doing that does exactly the same thing as removing an electron from the Valence band. 316 00:34:39,060 --> 00:34:47,040 So these are actually the same state. Okay. And, and so that's the price you have to pay for this sort of semiconductor picture of a superconductor. 317 00:34:47,610 --> 00:34:54,870 So what? So the interesting question then is, is could these things, you know, you know, sort of band structures, 318 00:34:54,870 --> 00:34:59,340 if you will, that have this intrinsic particle hall symmetry, could they have topological classes? 319 00:34:59,580 --> 00:35:03,150 And the way you can think about that is you could ask, well, what happens at the end? 320 00:35:03,480 --> 00:35:10,050 And at the end of a one dimensional, topological superconductor, it's possible to have a state that's at exactly zero energy. 321 00:35:10,770 --> 00:35:18,329 Okay. And if you have this state at zero energy, it's topologically protected because if it tried to move away from zero energy, 322 00:35:18,330 --> 00:35:21,750 then its partner would have to appear out of nowhere. And that can't happen. 323 00:35:22,320 --> 00:35:26,310 Okay. So. So this zero mode is topologically protected. 324 00:35:26,830 --> 00:35:28,950 Okay. And so. So. So you could ask, well, 325 00:35:28,950 --> 00:35:35,970 what under what circumstances will you get the zero mode where you'll get the zero mode at the end of a one dimensional topological superconductor? 326 00:35:36,420 --> 00:35:42,270 Okay. Now, this zero mode is actually a a very magical kind of thing, 327 00:35:42,790 --> 00:35:46,620 and it's something that we very, very much would like to make happen in the real world. 328 00:35:46,680 --> 00:35:50,520 Okay. Now what? So. So remember I told you that we have this redundancy? 329 00:35:50,520 --> 00:35:56,520 Adding a particle at energy is the same as removing a particle of energy minus E. 330 00:35:57,120 --> 00:36:02,760 So that means that adding a particle at zero energy is the same thing as removing a particle at zero energy. 331 00:36:03,720 --> 00:36:08,370 So that means that this particle is its own antiparticle. 332 00:36:08,880 --> 00:36:13,610 Okay, I put one in. I put another one, and poof, they're gone. Okay. 333 00:36:14,180 --> 00:36:18,259 So. So this is a particle which is its own anti particle. 334 00:36:18,260 --> 00:36:28,910 And so this is actually an example of a kind of particle that was introduced almost, almost a century ago by a Tory maia mana. 335 00:36:29,600 --> 00:36:36,590 So my R.A. was a I was a brilliant theoretical physicist, a protege of Enrico Fermi's, 336 00:36:37,400 --> 00:36:43,670 who discovered shortly after Dirac invented relativistic quantum mechanics, 337 00:36:43,940 --> 00:36:53,780 that that a consistent interpretation of Dirac's equation was that it's possible to have a particle that is its own antiparticle. 338 00:36:54,810 --> 00:37:02,610 Okay. Now, Maiorana had a tragically short career, and shortly after shortly after that discovery, he disappeared. 339 00:37:02,610 --> 00:37:05,130 And it's a big question what happened to him. Okay. 340 00:37:05,470 --> 00:37:16,140 But so so the minor on affirming on is something that that we very much would like to have now in in condensed matter physics. 341 00:37:16,140 --> 00:37:19,860 Of course we don't really have the fundamental majorana fermions is micron. 342 00:37:19,980 --> 00:37:25,200 So these majorana fermions are sort of emergent, if you will, maiorana fermions. 343 00:37:25,440 --> 00:37:28,319 But they're something we very much would like to make. 344 00:37:28,320 --> 00:37:38,040 And the reason we would like to make it is due to an idea from a modern day genius named Aleksey Katsav, 345 00:37:38,310 --> 00:37:41,940 who realised that if you could, if you could make these, 346 00:37:42,270 --> 00:37:47,190 these majorana bound states at the end of a topological superconductor, 347 00:37:47,430 --> 00:37:55,080 then they can give you a new method for storing and manipulating quantum information. 348 00:37:56,520 --> 00:38:04,470 And and so by making these, they could give you a route to making a kind of quantum computer. 349 00:38:04,690 --> 00:38:11,669 Okay. And now this is something we are not close to implementing yet. 350 00:38:11,670 --> 00:38:15,210 We're not ready to make a quantum computer out of these things. 351 00:38:15,450 --> 00:38:22,590 But this gives tremendous motivation for us to make this happen in the real world. 352 00:38:23,010 --> 00:38:30,690 Okay. And so so there's a huge effort going on in the world to try to discover these a mirror on a Fermi. 353 00:38:30,930 --> 00:38:35,550 This is a little bit of a little bit of a busy slide. And so I don't want to go through it in detail. 354 00:38:35,730 --> 00:38:44,250 I just want to say that there are various proposals for how one can implement it to realise these majorana zero modes. 355 00:38:44,250 --> 00:38:47,820 So these are, these are the majorana zero modes that one can hopefully make. 356 00:38:48,030 --> 00:38:55,170 And, and there is very encouraging evidence that these majorana zero modes may have already been discovered. 357 00:38:55,320 --> 00:38:59,880 Okay. So the challenge is to make sure that these experiments are really right. 358 00:39:00,540 --> 00:39:06,569 And there's there's there's huge debates going on as we speak about these things. 359 00:39:06,570 --> 00:39:16,200 But but this is certainly something which has galvanised our, our community, the idea that we can create these majorana zero modes in the laboratory. 360 00:39:16,740 --> 00:39:23,760 Okay. So what I'd like to do in the remaining time is to shift gears completely. 361 00:39:23,760 --> 00:39:34,430 So what I hope I've convinced you is that this idea of topology and topologically protected boundary modes is sort of a ubiquitous idea, 362 00:39:34,970 --> 00:39:41,340 and it shows up in sort of many different contexts in this electronic quantum electronic states of matter. 363 00:39:42,660 --> 00:39:48,540 And so what I want to show you now is how this very similar idea shows up in a completely different situation. 364 00:39:48,660 --> 00:39:54,180 Okay. And so the set of problems that I'm going to describe, you are actually even older than the problems of quantum mechanics. 365 00:39:54,180 --> 00:40:00,480 They predate quantum mechanics itself. So these are a set of problems that go back to the days of of Maxwell. 366 00:40:00,960 --> 00:40:11,910 Okay. So so now, of course, many of you will have heard of Maxwell as the Maxwell who created the the laws of electromagnetism. 367 00:40:13,410 --> 00:40:16,620 But Maxwell also had other interesting interests as well. 368 00:40:16,620 --> 00:40:21,570 And one of them was the interest of sort of what you might call structural engineering. 369 00:40:21,870 --> 00:40:29,500 Okay. The question being, you know, if you make some sort of frame or configuration of mass is the question is, is it mechanically stable? 370 00:40:29,520 --> 00:40:36,320 So so imagine you're trying to build a house and you build a house out of, you know, bars and springs like this. 371 00:40:36,330 --> 00:40:39,780 You know, if you build it like this, you might worry that something like that would happen. 372 00:40:40,290 --> 00:40:43,849 Okay. And then your house would fall down. Okay. So. 373 00:40:43,850 --> 00:40:48,570 So what's wrong here is that there's a way of there's a there's a degree of motion 374 00:40:48,840 --> 00:40:54,060 that doesn't have any restoring force that doesn't that that that that is floppy, 375 00:40:54,150 --> 00:41:02,959 if you will. Okay. And so what Maxwell was interested in is he was interested if you have some configuration of masses connected by springs, 376 00:41:02,960 --> 00:41:06,360 say, then how many of these floppy modes are there? 377 00:41:07,230 --> 00:41:10,230 And so the insight he had was that, well, 378 00:41:10,260 --> 00:41:17,640 what you need to do is you need to count the number of degrees of freedom in your system, which in which case, you know. 379 00:41:17,730 --> 00:41:21,570 So in two dimensions each, each, each mass can move in two directions. 380 00:41:21,570 --> 00:41:24,000 So each mass has two degrees of freedom. Okay. 381 00:41:24,240 --> 00:41:29,970 And then you need to compare that with the number of constraints that you have, which is like the number of springs. 382 00:41:30,810 --> 00:41:36,030 And so what Maxwell showed is that taking the difference between the number of degrees of 383 00:41:36,030 --> 00:41:42,240 freedom and the number of constraints gives you a lower bound for the number of floppy amounts. 384 00:41:42,630 --> 00:41:47,010 So if you don't have enough springs, then, then, then then your house will fall down. 385 00:41:47,490 --> 00:41:54,270 Okay. Now, of course. So what you might do, I, I, you know, if you wanted to make. 386 00:41:54,340 --> 00:41:58,600 A better house is you'd probably put some more springs in and sort of stiffen it up. 387 00:41:58,600 --> 00:42:05,950 And if you did that and then you got rid of these floppy motions now, but of course you could put as many springs than you want. 388 00:42:06,160 --> 00:42:13,420 So Maxwell's rule is not an equality, okay, because I could have lots and lots of springs, lots and lots of bonds. 389 00:42:14,320 --> 00:42:27,340 So Maxwell's rule is an inequality. Okay, now, so this is actually a problem that is of interest in the Structural Engineering Committee. 390 00:42:27,340 --> 00:42:35,040 And so the engineer has actually revisited this problem and realised that when you put these cross bonds on, you actually get something new, okay? 391 00:42:36,280 --> 00:42:46,120 And what you get is you get a configuration where the bonds are redundant in the following sense that I could take these bonds if I, 392 00:42:46,120 --> 00:42:50,589 if I, if I tightened the Red Springs. So the Red Springs are under tension. 393 00:42:50,590 --> 00:42:56,050 They're pulling in and I loosened the Green Springs so that they're under compression. 394 00:42:56,200 --> 00:43:03,450 They're pushing out. Then the forces on all of the masses is equal to zero. 395 00:43:03,480 --> 00:43:08,090 They can those forces can cancel each other. And so this is an equilibrium situation. 396 00:43:08,100 --> 00:43:12,540 There are no forces. So this is what is called a state of self stress. 397 00:43:13,050 --> 00:43:21,870 Okay. And what. And so Khalidi is it was an engineer who in the 1970s revisited this problem. 398 00:43:22,080 --> 00:43:28,740 And what he was able to show is that Maxwell's counting rule can be turned into an equality. 399 00:43:29,590 --> 00:43:33,419 If you take into account the state of self stress rates of self stress. 400 00:43:33,420 --> 00:43:36,610 So the difference between the number of floppy modes and the number of these, 401 00:43:36,610 --> 00:43:44,020 the number of independent ways you can do this is equal to the difference between the number of degrees of freedom and the number of constraints. 402 00:43:44,710 --> 00:43:49,220 Okay. So this is this is the Maxwell problem. And so so so what? 403 00:43:49,240 --> 00:43:52,270 So where I entered this, thinking about this. 404 00:43:52,270 --> 00:44:01,929 So my colleague Tom Lenski, who is a so maybe some of you know Tom, he's one of the pioneers of soft condensed matter physics. 405 00:44:01,930 --> 00:44:05,340 And he's my colleague at the University of Pennsylvania. And he's been interested in these problems. 406 00:44:05,530 --> 00:44:15,189 And he was interested in the situation where where you have a balance between the number of of of of of degrees of freedom and number of constraints. 407 00:44:15,190 --> 00:44:21,160 So these would be situations where your system is just on the verge of mechanical instability. 408 00:44:21,790 --> 00:44:31,749 And so so in particular, the the the set of problems that he introduced me to where the idea of having a periodic what's called ISOs static lattice. 409 00:44:31,750 --> 00:44:42,190 So ISO static means that that the number of of of sites and bonds are such that this count is equal to zero. 410 00:44:43,210 --> 00:44:46,960 Okay. And so, for example, a square lattice is like that. 411 00:44:46,960 --> 00:44:53,410 Each site has two degrees of freedom and then there are sort of twice as many bonds as there are sites. 412 00:44:54,280 --> 00:45:01,630 This lattice is called a kagome, a lattice, and that also has the same property in three dimensions. 413 00:45:02,350 --> 00:45:05,830 There's a lattice called up hydrochloride lattice that has this property. 414 00:45:06,130 --> 00:45:13,360 Okay. And so, so so these are this is so so the interesting question is what happens to these lattices? 415 00:45:13,360 --> 00:45:19,179 And and so so my colleague Tom, he sort of viewed this as a model system for studying a number of kinds of problems that 416 00:45:19,180 --> 00:45:24,820 are of interest to soft matter physicists and statistical physicists and also engineers, 417 00:45:25,300 --> 00:45:28,930 problems like rigidity, percolation jamming, network glasses. 418 00:45:28,930 --> 00:45:34,420 And so these are these are the set of sort of physics problems that these models can be applied to. 419 00:45:34,750 --> 00:45:44,020 Okay. And so so when Tom thought about this, he realised that there's an interesting situation. 420 00:45:44,450 --> 00:45:49,179 And the interesting situation is to think about this, this category, a lattice. 421 00:45:49,180 --> 00:45:54,879 And so here's what Tom showed. So, so if you if you if you just have one of these cockamamie lattices. 422 00:45:54,880 --> 00:46:04,690 So, again, what I want you to imagine is that at every point here, there's a mass and every line there's a spring connecting those masses. 423 00:46:04,840 --> 00:46:12,130 Okay, so it's sort of a mass and spring problem. It's the kind of problem you could almost imagine giving a sophomore physics student, 424 00:46:12,430 --> 00:46:15,550 you know, figure out what what the motion of this kind of thing would be. 425 00:46:15,850 --> 00:46:22,430 Okay. So so one thing you can see is if you have a lattice like this, it has it has lots of floppy moats. 426 00:46:23,130 --> 00:46:27,940 Okay. So they, you know, the things can sort of go like this and there's no restoring force for that. 427 00:46:28,270 --> 00:46:34,510 Okay. Now, there are also lots of states of self stress. 428 00:46:34,900 --> 00:46:41,350 So so if I have a straight line of bonds like this, then, you know, 429 00:46:41,440 --> 00:46:46,060 if I put all of these bonds under tension, then then all the forces are going to cancel out. 430 00:46:46,720 --> 00:46:51,250 Okay. And so, so that makes sense. You have lots of floppy modes, lots of states have self stress. 431 00:46:51,250 --> 00:46:54,309 So this count, you know, so it's perfectly static. 432 00:46:54,310 --> 00:46:57,310 So that means the number of these has to be equal to the number of these. So that's fine. 433 00:46:57,790 --> 00:46:59,020 Okay. There are lots of both of them. 434 00:47:00,390 --> 00:47:09,060 But what Tom realised is that he can do a very simple modification of this kagome lattice by twisting the triangle. 435 00:47:09,060 --> 00:47:15,900 So I'm going to rotate the triangles, the rotate the up pointing triangles to the left and the down pointing triangles to the right. 436 00:47:16,170 --> 00:47:22,139 And by doing that, you can sort of see that you get rid of the states of self stress, because if I have these zigzag bars, 437 00:47:22,140 --> 00:47:28,590 if I put these all under tension, then there's going to be a net force, you know, in the vertical direction on on the sides. 438 00:47:28,590 --> 00:47:30,630 So that's not a state of self stress anymore. 439 00:47:30,930 --> 00:47:41,030 So, in fact, by doing this modification, you get rid of of all of the states of self stress and the states and the and the and the floppy modes. 440 00:47:41,070 --> 00:47:44,130 Okay. And so this so the system sort of stiffens up. Okay. 441 00:47:44,880 --> 00:47:46,709 But the thing that's remarkable about this, 442 00:47:46,710 --> 00:47:54,840 so this this is what happens if you imagine sort of wrapping it periodically around so that there's no boundary. 443 00:47:55,560 --> 00:48:01,100 What Tom realised is that when there is a boundary, something very interesting happens, okay? 444 00:48:01,890 --> 00:48:07,740 And in particular, so again, this is this is a problem that one could imagine for a second year physics student, 445 00:48:09,030 --> 00:48:12,510 you know, solving for the motion of these masses and springs. 446 00:48:12,690 --> 00:48:18,180 And what Tom discovered is that when he cuts open this lattice so that there's a free boundary, 447 00:48:18,480 --> 00:48:29,040 then even though we got rid of all the zero modes in the bulk, there are floppy modes localised on the boundary. 448 00:48:29,890 --> 00:48:38,650 Okay. And so. So the way a physicist would analyse this problem is by looking at the sort of natural 449 00:48:38,650 --> 00:48:46,870 frequencies of vibration of this system as a function of the sort of wavelength in the, 450 00:48:47,110 --> 00:48:52,000 in the X direction. And. And so what one, so want to have a picture like this of the normal nodes. 451 00:48:52,150 --> 00:49:00,280 And the important thing here is that there are all of these zero frequency modes. 452 00:49:01,810 --> 00:49:09,640 Okay. And now. So. So actually, before Tom did this calculation, he knew that these zero modes had to be there. 453 00:49:09,760 --> 00:49:18,100 Okay. He had an ingenious argument based on this Maxwell accounting rule that that that guarantees that these zero modes have to be there. 454 00:49:18,670 --> 00:49:23,049 But once you look at these, you know, I mean, so so you can change things. 455 00:49:23,050 --> 00:49:27,430 You can move the triangles a little bit. And these zero modes don't go away. 456 00:49:27,910 --> 00:49:31,360 They're stuck there. It's like their topological. 457 00:49:32,360 --> 00:49:37,640 Okay. So now so Tom, he, you know, his office is down the hall from mine. 458 00:49:38,720 --> 00:49:44,570 And so we, you know, we work in very different fields, so we don't often collaborate together, but we have written papers together in the past. 459 00:49:44,570 --> 00:49:50,000 And so we bump into each other in the hallway and we talk and and and so so one day, 460 00:49:50,000 --> 00:49:55,250 a couple of years ago, you know, Tom came up to me and he said, you know, you know. 461 00:49:57,100 --> 00:50:04,690 Are my boundary moats related to your boundary moats? And so I said, no, Tom, no. 462 00:50:05,380 --> 00:50:12,130 You know, I you know, I'm I'm doing, you know, topological quantum, you know, states of matter. 463 00:50:12,130 --> 00:50:20,770 And, you know, you're doing like tinker toys, right? So so I, you know, so I was not very receptive to this at first. 464 00:50:20,770 --> 00:50:28,810 But Tom, you know, he's a he actually understood some things that I didn't understand and he was persistent. 465 00:50:29,080 --> 00:50:33,209 And so he came back. He kept coming back. Came back to me again. 466 00:50:33,210 --> 00:50:37,290 So finally I agreed to sit down with him and talk. 467 00:50:37,380 --> 00:50:41,010 Talk about it in his office for a while, tend to think more seriously about it. 468 00:50:41,010 --> 00:50:45,580 And so what I decided to do, I started to come up with the argument that would put this to rest. 469 00:50:46,020 --> 00:50:54,240 Okay. And so here is the argument that I constructed. I said, Tom, look, I'm doing quantum mechanics, okay? 470 00:50:54,510 --> 00:51:02,250 So quantum mechanics is based on the Schrödinger equation, which is a first order differential equation in time. 471 00:51:04,170 --> 00:51:11,070 And what one is interested in solving the Schrödinger equation is one is interested in solving for the spectrum of the Hamiltonian, 472 00:51:11,220 --> 00:51:15,360 which are what are called the eigenvalues of this Hamiltonian matrix. 473 00:51:15,540 --> 00:51:23,999 And the important thing for me is that the eigen, the Hamiltonian has positive and negative energy eigenvalues, 474 00:51:24,000 --> 00:51:27,990 which form the conduction band and the valence band. 475 00:51:28,480 --> 00:51:36,570 Okay. And all of my topological band theory is all about Topologically classifying the valence band. 476 00:51:37,580 --> 00:51:40,790 So that's what I'm about. Okay. But Tom. 477 00:51:41,660 --> 00:51:51,110 Tom is just solving Newton's laws. F equals M-A and Newton's laws are second order equations in time. 478 00:51:51,500 --> 00:51:57,770 Okay, so now, of course, there's still you know, if you want to figure out what the normal modes of vibration of a system are, 479 00:51:57,770 --> 00:52:00,799 you have a, you have a a matrix, two diagonals. 480 00:52:00,800 --> 00:52:04,970 You want to find the spectrum of of this what's called dynamical matrix. 481 00:52:05,660 --> 00:52:09,940 But this dynamical matrix is very different than the Hamiltonian in the sense that it's 482 00:52:09,980 --> 00:52:17,299 eigenvalues are all positive and then the frequency squared is equal to the eigenvalues of this, 483 00:52:17,300 --> 00:52:27,110 of this dynamical matrix. And so, so, so Tom doesn't have a valence band, you know, the, you know, the normal modes of vibration. 484 00:52:27,110 --> 00:52:32,350 The frequency is always positive. So I said, Tom, you know, these problems are completely different. 485 00:52:33,630 --> 00:52:39,520 But then. When I said it that way. I was reminded of a story. 486 00:52:42,080 --> 00:52:49,340 And. This is the story of the legendary Solvay Conference of 1927. 487 00:52:49,340 --> 00:52:58,250 So this is actually one of my favourite pictures. So, you know, as a physicist, you recognise a lot of faces in this in this Line-Up here. 488 00:52:58,790 --> 00:53:02,270 But for me at least there there are a number of faces that I didn't recognise. 489 00:53:02,900 --> 00:53:05,360 But what was remarkable is that when I went and looked them up, 490 00:53:06,140 --> 00:53:15,050 I realised that I knew all of their names because their names sort of live on in the nomenklatura of of our field. 491 00:53:15,830 --> 00:53:18,469 And so this is this is actually a remarkable thing. So, so in that case, 492 00:53:18,470 --> 00:53:26,570 this Solvay conference was a conference that brought together the pioneers of quantum theory to debate that sort of new emerging subject. 493 00:53:26,570 --> 00:53:32,420 So in particular, Niels Bohr, you know, the famous grandfather of quantum theory, 494 00:53:32,810 --> 00:53:41,900 was there, and also Paul Dirac, who was young and not yet quite so famous. 495 00:53:42,380 --> 00:53:50,620 Okay. So the story goes that that Bohr bumps into Dirac during a coffee break and asked him what he's up to. 496 00:53:51,080 --> 00:54:01,640 And so Dirac was a man of notoriously few words, and he said, I'm trying to take the square root of something. 497 00:54:05,160 --> 00:54:08,980 So Dirac. Trying to take the square root. 498 00:54:09,010 --> 00:54:13,780 Niels Bohr was okay. You work on that? 499 00:54:17,440 --> 00:54:28,930 What we now know is that the square root that Dirac was trying to take is probably the most consequential square root in the history of physics, 500 00:54:29,770 --> 00:54:40,750 because Dirac was trying to take the square root of the Kleine Gordin equation to unify quantum mechanics with the theory of relativity. 501 00:54:42,160 --> 00:54:47,950 And Dirac came up with a ingenious way. 502 00:54:48,520 --> 00:54:53,300 So the question is, how do you take the square root of this without taking a square root? 503 00:54:53,320 --> 00:55:01,750 And Dirac came up with an ingenious way to do it by turning it again into a matrix and multiplying the two matrices together. 504 00:55:03,720 --> 00:55:07,980 And in doing so he discovered the Dirac equation. 505 00:55:09,490 --> 00:55:17,590 Now the direct equation, in addition to explaining the microscopic origin of the spin of the electron. 506 00:55:18,980 --> 00:55:23,510 You made another prediction, which is that the electrons have a valence band. 507 00:55:25,140 --> 00:55:28,920 Which is the which gives us the antiparticle of the electron. 508 00:55:29,440 --> 00:55:33,720 Okay. So, so, so, so. So Dirac discovered the positron. 509 00:55:33,930 --> 00:55:38,830 But the positron is really just the valence band. Okay. 510 00:55:39,340 --> 00:55:50,670 So I thought to myself. If I could just take the square root of Tom's dynamical matrix, then maybe I could be like Dirac. 511 00:55:51,210 --> 00:55:57,210 So, okay, so I went home. So I, so I went home and slept on this and I realised that you can take the square root. 512 00:55:57,450 --> 00:56:03,780 Now I have to apologise. This is a little bit of a technical discussion here. 513 00:56:05,160 --> 00:56:07,799 And I promise you, this is the only technical discussion. 514 00:56:07,800 --> 00:56:13,890 But but, but it's so cool that I just for the physicists in the audience, I want to take you through this briefly. 515 00:56:13,990 --> 00:56:17,610 Okay. So so the idea here is the following. 516 00:56:17,610 --> 00:56:25,710 So the question is, so this dynamical matrix has a very special form because, you know, really. 517 00:56:25,980 --> 00:56:31,050 So the what I want you to think about is think about what the energy of this system of masses and springs are. 518 00:56:31,530 --> 00:56:37,620 So the, the energy of a spring is just one half k x squared is just hooks, hooks law for the springs. 519 00:56:38,310 --> 00:56:51,000 But are the the compression or the extension of a spring is related to the displacement of the sites of the spring by some linear operator. 520 00:56:51,630 --> 00:56:59,040 So each spring, the amount I stretch each spring by depends on the extension of the springs by a q. 521 00:56:59,190 --> 00:57:03,030 And what that means is that the energy is q. Q transpose. 522 00:57:04,940 --> 00:57:09,850 Okay. So it's almost. So the dynamical matrix is almost a perfect square. 523 00:57:09,860 --> 00:57:20,120 It's not quite it's not cu squared, it's cu times to transpose. But this form sort of lights, another light bulb in your head, which is supersymmetry. 524 00:57:20,750 --> 00:57:24,650 This has exactly the same structure as supersymmetric quantum mechanics. 525 00:57:24,860 --> 00:57:32,270 And so it turns out this matrix has a partner that looks like this that has exactly the same spectrum except for the zero votes. 526 00:57:33,110 --> 00:57:36,110 Okay. And to cut a long story short. 527 00:57:36,290 --> 00:57:46,490 One can combine these two matrices into these these two operators into a bigger matrix, sort of like what Dirac did. 528 00:57:47,570 --> 00:57:51,230 Okay. And then this thing is a perfect square. 529 00:57:53,180 --> 00:57:58,760 Okay. And so now the thing that I so now I have taken the square root of Tom. 530 00:58:00,010 --> 00:58:05,050 And now I have a Hamiltonian that I can interpret as a quantum Hamiltonian. 531 00:58:05,940 --> 00:58:13,259 Okay. And so once I have this, then I can apply everything that I know about topological band theory. 532 00:58:13,260 --> 00:58:16,320 So this one has a valence band. Okay. And. 533 00:58:16,800 --> 00:58:21,930 And so. So just if there are any aficionados in the audience, the thing you have to do is understand the symmetries. 534 00:58:22,140 --> 00:58:29,600 There's. There's a time reversal symmetry in a particle hole symmetry. And and in in the lingo that we have, this is called a class B one. 535 00:58:29,610 --> 00:58:34,080 So this is something that I completely recognised anew. Once I knew this, I knew how to proceed. 536 00:58:34,770 --> 00:58:36,570 Okay. And so by doing this, 537 00:58:37,350 --> 00:58:47,820 one of the things we were able to discover is that even when you don't have any imbalance between the number of sites and the number of bonds, 538 00:58:48,510 --> 00:58:55,230 you can have floppy modes if you have an interface between two topologically distinct regions. 539 00:58:56,160 --> 00:59:02,850 Okay. And so, so. So this is an example where if I was just to count the sites in the bonds, I would say that there shouldn't be anything. 540 00:59:02,940 --> 00:59:08,760 But now I have localised floppy modes and then there are also localised states of self stress on this boundary. 541 00:59:09,360 --> 00:59:15,540 Okay. So now just in the last few minutes, I want to I'm not going to go through this technical stuff, 542 00:59:15,690 --> 00:59:19,560 but I want to show you this just because it's kind of fun. 543 00:59:19,920 --> 00:59:30,360 Okay. And so so, you know, once I had this mapping between the sort of classical mechanics problem and the quantum electronics problem, 544 00:59:30,360 --> 00:59:36,660 then, you know, I told you that my strategy in these things is always to ask, well, what's the simplest example of of this? 545 00:59:37,050 --> 00:59:40,110 Okay. What's the simplest example of a topological phenomenon? 546 00:59:40,320 --> 00:59:45,660 And so the simplest example in topological band theory was, I already told you it's the sushi for here tomorrow. 547 00:59:45,660 --> 00:59:49,110 Is this one dimensional, one dimensional polymer. Okay. 548 00:59:49,110 --> 00:59:53,700 And so, so so this poses the question, what's the mechanical analogue of that? 549 00:59:54,560 --> 01:00:01,230 And so, so once I realised that this was a question to ask, then it was a very pleasurable evening to sit down and work out what it was. 550 01:00:01,320 --> 01:00:08,140 Here's, here's, here's what I came up with. So, so, so I want you to imagine a configuration where I have a bar. 551 01:00:08,160 --> 01:00:12,030 It's a rigid bar. And then these blue masses can rotate around pivots. 552 01:00:12,600 --> 01:00:16,770 Okay. And then they're connected together by springs. Okay. 553 01:00:16,920 --> 01:00:21,930 So in this case, you can imagine that you could push this and everything would all the blue things would move to the right. 554 01:00:22,680 --> 01:00:26,010 And that would that would be sort of a floppy motion. 555 01:00:26,220 --> 01:00:31,620 This is the analogue of the metallic state of the sushi for Jaeger model. 556 01:00:31,920 --> 01:00:45,380 Okay. But now what I can do is I can shorten the the springs in such a way that, that, that the the the the the bars want to rotate. 557 01:00:45,390 --> 01:00:52,670 So now I have them set up like this. And so now you can see that this guy can't move without stretching a sprint. 558 01:00:53,350 --> 01:00:56,360 Okay. This guy can't move without stretching a spring either. 559 01:00:56,750 --> 01:01:01,820 Okay, so these so these are the mechanical modes. 560 01:01:01,910 --> 01:01:05,270 All have an energy gap for for for these motions. 561 01:01:05,750 --> 01:01:08,800 But look at this guy. At the end, he can move. Okay. 562 01:01:09,050 --> 01:01:14,150 That's your zero. Now, of course, you might say, well, this is an artefact of them being parallel. 563 01:01:14,150 --> 01:01:17,330 If you if you do it like this, then actually what you find is there's still a zero remote. 564 01:01:18,020 --> 01:01:23,990 If you have an interface now between one going to the left and one going to the right, there's a zero mode there. 565 01:01:24,500 --> 01:01:30,079 Now, the reason I want to show you this is because know, you know, so I came up with this idea, I drew the picture or I wrote the paper, 566 01:01:30,080 --> 01:01:34,940 you know, and and I was always thinking to myself, you know, gosh, wouldn't it be nice to make one of these toys to play with? 567 01:01:35,840 --> 01:01:40,129 And but I never really, you know, I don't know, had the energy to sit down and do it. 568 01:01:40,130 --> 01:01:46,400 But. But my friend Vincenzo Vitale, who's a professor at the University of Leiden, he got very excited about this. 569 01:01:46,640 --> 01:01:52,860 And so he got so excited that he convinced his boss to to give him a laboratory. 570 01:01:52,880 --> 01:01:56,660 He's a theorist, but but he convinced his boss to give him a laboratory, 571 01:01:56,780 --> 01:02:02,450 which he has stocked full of Lego to be able to put together and make all these kinds of mechanical things. 572 01:02:02,570 --> 01:02:08,630 And so I went and visited him in Leiden, and he very generously gave me one of his toys. 573 01:02:10,010 --> 01:02:13,310 And so so let's see if I can do this here. 574 01:02:16,400 --> 01:02:19,460 All right. So this is the. 575 01:02:21,330 --> 01:02:24,620 Is the. Is what I have here. 576 01:02:26,150 --> 01:02:35,840 Okay. And so what I hope you can see. So so now the only difference I've made is that the Red Springs I have replaced by these orange bar. 577 01:02:35,860 --> 01:02:38,960 So it's like the springs are infinitely stiff. Okay. 578 01:02:39,140 --> 01:02:43,219 So what you can see is that indeed everything is sort of, you know, locked here. 579 01:02:43,220 --> 01:02:46,250 It can't can't move without doing it, but. 580 01:02:47,700 --> 01:02:52,710 There it is. So. So this is like a very low tech micron to zero mode. 581 01:02:54,210 --> 01:03:00,870 Okay. And so so so this really is a topological boundary mode. 582 01:03:01,500 --> 01:03:05,910 Okay. And, you know, it can exist. 583 01:03:06,210 --> 01:03:09,720 You know, so so now I have a situation. I moved it. So I have things. 584 01:03:09,930 --> 01:03:14,610 Everything is rigid here and rigid here. But I have a boundary mode in between them. 585 01:03:15,680 --> 01:03:19,790 Okay. You would never discover this boundary mode by counting sight. 586 01:03:20,120 --> 01:03:27,560 You know, you never discover where it is by counting locally the number of degrees of freedom in the number of constraints. 587 01:03:27,710 --> 01:03:34,070 Okay. Now, there's actually one thing that, you know, I didn't you know, as I said, I didn't have the energy to actually make one of these things. 588 01:03:34,070 --> 01:03:43,330 And and actually, I kind of regret it, because when Vincenzo made this, he actually discovered something that we didn't realise, okay. 589 01:03:43,580 --> 01:03:48,580 Which is that, you know, so one question you could ask is, well, what happens if you go beyond the linear regime? 590 01:03:48,590 --> 01:03:52,640 You know, what happens if you keep pushing the zero mode? Well, if you keep pushing this zero mode, then. 591 01:03:56,220 --> 01:04:00,860 It goes all the way down to the other side. So this thing, it's almost like a particle, right? 592 01:04:02,530 --> 01:04:10,149 Okay. So. So there's actually more kind of interesting nonlinear physics that goes on in this and this in this kind of structure as well. 593 01:04:10,150 --> 01:04:16,720 And so so this is something that, you know, is a subject for for for for more investigation. 594 01:04:16,730 --> 01:04:19,300 Okay. So I think what I want to do is I want to finish. 595 01:04:19,660 --> 01:04:28,149 And so the main lesson that I wanted to get across here was that this phenomenon of of topology and topological 596 01:04:28,150 --> 01:04:34,660 boundary modes is this really is sort of an elegant mathematical construction and it has applications in, 597 01:04:34,810 --> 01:04:43,600 in, in very diverse venues, the topological electronic phenomena as well as mechanical modes in these ISO static systems. 598 01:04:43,870 --> 01:04:53,380 So, so we have a lot more things to do. So of course, you know, my main home is in the sort of electronic materials and, and, you know, 599 01:04:53,530 --> 01:04:58,150 we want to study new materials, new phenomena, new experiments in these quantum electronic systems. 600 01:04:58,750 --> 01:05:04,299 But there's also many interesting questions in the in the mechanical systems. 601 01:05:04,300 --> 01:05:13,450 You know, so experiments on metamaterials and maybe there are optical electronic plasmonic versions of this kind of phenomena as well. 602 01:05:13,600 --> 01:05:20,440 And of course, an interesting question in both of these settings is the role of interactions and nonlinear narratives. 603 01:05:20,440 --> 01:05:24,490 And those are going to be things which I think are going to keep us busy for for some time to come. 604 01:05:24,940 --> 01:05:25,750 So thank you very much.