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 SchrÃ¶dinger 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 SchrÃ¶dinger 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.