1
00:00:10,990 --> 00:00:16,990
So for our next story we have Professor Sid Parameswaran who is a condensed,

2
00:00:16,990 --> 00:00:22,840
massive theoretical physicist, and Sid is going to tell us about axioms in the solid state.

3
00:00:24,830 --> 00:00:29,480
Thank you, Julia, and thank you all for being here. It's really nice to be back for one of these events in person.

4
00:00:29,810 --> 00:00:35,150
So I should confess that I made the stop this week. I have an 18 month old son and I tried it on him a couple of days ago.

5
00:00:35,360 --> 00:00:40,100
He promptly fell asleep, which I wasn't complaining about, but I'm hoping for a different outcome today.

6
00:00:40,670 --> 00:00:47,600
So in Joe's stock, what he mentioned was that he talked to you about some of the exciting physics of actions and action like particles.

7
00:00:47,930 --> 00:00:52,430
And so what I want to do in this stock is actually illustrate how some of those ideas apply in a setting,

8
00:00:52,430 --> 00:00:57,380
perhaps where we're more familiar with in everyday physics is sort of an introductory courses,

9
00:00:57,620 --> 00:01:04,590
which is Maxwell electrodynamics, but with an ingredient, which is what do these axioms mean for thinking about electrodynamics?

10
00:01:04,610 --> 00:01:10,790
So that's what I want to do today. So I want to start with a pair of propositions and axioms from this paper by Frank.

11
00:01:10,790 --> 00:01:15,679
Well, check. And you know, he famously coined the phrase Axion in its original context.

12
00:01:15,680 --> 00:01:18,889
And quickly, Jordan mentioned this sort of running joke on axioms.

13
00:01:18,890 --> 00:01:25,430
The word it sounds like a fancy particle was actually a form of laundry soap because it cleaned up problems in the standard model.

14
00:01:25,430 --> 00:01:33,530
And so that was why the name came from. So the first quote expresses a sentiment that many of us in the centre can probably sympathise with,

15
00:01:33,860 --> 00:01:39,380
which is that whether or not axioms have any physical reality, that study can be a useful intellectual exercise.

16
00:01:39,470 --> 00:01:44,660
So that's great. That keeps us employed. The second point, he said, and that's the centre of the stock,

17
00:01:44,870 --> 00:01:52,670
is that it might be is not beyond the realm of possibility that fields whose properties partially mimic those of Axion fields,

18
00:01:52,880 --> 00:01:58,910
can be realised in condensed matter systems. So it's the second proposition that's going to be my primary concern in the stock.

19
00:01:59,240 --> 00:02:06,980
And the first is something though I'll return to at the end. So to start off with, let's just say let's set up rules of the game.

20
00:02:07,220 --> 00:02:13,010
So what Joe talked about in high energy physics and what John will talk about is the Axion field theta as

21
00:02:13,010 --> 00:02:17,870
a fundamental field that describes Axion dynamics in quantum chroma dynamics and its original setting.

22
00:02:18,410 --> 00:02:24,560
This is a challenge to with I stated goal of finding Axion like physics and solids because we know the standard model of solids.

23
00:02:24,830 --> 00:02:30,740
The standard model of solids is just electrons and ions governed by Maxwell's equations, interacting by Coulomb forces.

24
00:02:31,190 --> 00:02:34,280
And so there's no QCD here, there's no strong force.

25
00:02:34,520 --> 00:02:40,640
And so very clearly I can't change this description when I'm talking about solids, the sort of ultraviolet physics is fixed.

26
00:02:41,090 --> 00:02:45,260
And so I want to think about axioms. And so my question is, how can I do that in a solid.

27
00:02:46,130 --> 00:02:50,650
So of course, if you've been on Saturday Theory before and seen a condensed matter session, you sort of know the answer.

28
00:02:50,960 --> 00:02:54,050
That instead of looking at the physics of the high energy scales,

29
00:02:54,050 --> 00:02:58,400
we look at low energy scales below at some very low scale that we can probe and experiments

30
00:02:58,640 --> 00:03:02,720
and ask for new physics that are emergent when many particles cooperate on those scales.

31
00:03:03,290 --> 00:03:06,680
So the goal of this talk is to find a way to obtain.

32
00:03:07,130 --> 00:03:12,480
So what I've written down is the Maxwell Lagrangian. So this is just ordinary electromagnetism, so nothing exotic of it.

33
00:03:12,560 --> 00:03:16,790
So this is very similar to what Joe talked about when he talked about Axion like particles.

34
00:03:17,090 --> 00:03:23,989
So the Maxwell Lagrangian with the term that comes from Axion electrodynamics and the disclaimer is that for most of the stock,

35
00:03:23,990 --> 00:03:27,090
I'm just going to think of this theta as an angle rather than a field.

36
00:03:27,090 --> 00:03:31,910
It's just going to be a constant. But I want to talk about what that constant means if we add that to our equations.

37
00:03:32,290 --> 00:03:36,800
Okay, so that's what I'm going to do today. So. As a warm up.

38
00:03:36,920 --> 00:03:39,860
I want to start off by reviewing a little bit about Maxwell's equation.

39
00:03:39,880 --> 00:03:44,780
So, you know, Maxwell's equations are sort of things we know about and love in vacuum,

40
00:03:44,990 --> 00:03:48,710
but I'm going to be interested because of my solid state physicists I care about matter.

41
00:03:48,950 --> 00:03:51,200
So I'm interested in Maxwell's equations in media.

42
00:03:51,440 --> 00:03:56,420
So this is part of the dreaded second year course where you have to deal with all kinds of terrible things for those.

43
00:03:56,630 --> 00:04:02,630
So trigger warning, I'm going to take you back to those days. So, you know, so those of you need to leave the room, you might want to do that now.

44
00:04:03,140 --> 00:04:06,530
So matter is a source of energy.

45
00:04:06,770 --> 00:04:11,419
We know that charges and currents and solids associated B and B and two of the Maxwell's equations.

46
00:04:11,420 --> 00:04:17,510
Oops. Then get there are two of the Maxwell's equations don't really care about the fact that they're sources of energy.

47
00:04:17,720 --> 00:04:21,980
So those are the fact that there are no magnetic monopoles, although we'll come back to that later.

48
00:04:22,280 --> 00:04:25,790
And the second is that there are that the electric and magnetic fields are linked.

49
00:04:26,390 --> 00:04:32,120
But do equations do care? Gauss's law and you know, the curl of B the currents generate fields.

50
00:04:32,120 --> 00:04:34,010
So Faraday's laws, they care about fields.

51
00:04:34,550 --> 00:04:42,379
And so to go any further, we have to actually model what we mean by charge and what the RO and g density and currents, what they mean.

52
00:04:42,380 --> 00:04:46,520
Because depending on whether we think of the system as a metal or an insulator, the behaviour will be different.

53
00:04:46,790 --> 00:04:54,409
So today I'm going to just talk about insulators where charges on electrons are fixed by positive and negative charges can't move very far apart.

54
00:04:54,410 --> 00:04:59,480
They're tied to each other and similarly currents don't roam freely, they're tied to sites of individual atoms.

55
00:05:00,080 --> 00:05:03,380
So if I draw a picture of a solid in a field, an insulator in a field,

56
00:05:03,650 --> 00:05:08,570
then it's just a bunch of little dipoles because every positive charge is pretty close to every negative charge.

57
00:05:08,930 --> 00:05:12,830
And if I looked at the current loops inside, they're all being fixed positions in space.

58
00:05:13,310 --> 00:05:17,270
And if I dropped you into this insulator and I walked around and I could measure electrical magnetic fields,

59
00:05:17,450 --> 00:05:23,240
you'd see wildly fluctuating electric and magnetic fields, because every time you went past a positive charge, things would jump up and down.

60
00:05:23,780 --> 00:05:28,909
So of course, this is very hard for us to deal with and compute, but what you can do is blow your eyes a little bit.

61
00:05:28,910 --> 00:05:34,010
That's very easy for me. That's all I need to do, you know? And what I do then.

62
00:05:34,010 --> 00:05:39,139
And there's a mathematical, mathematical statement behind that blurring, which is, of course, screening.

63
00:05:39,140 --> 00:05:46,010
All that means as we average over distances that are small compared to the size of the solid, but they're large compared to atomic distances.

64
00:05:46,370 --> 00:05:52,999
And if I do that, I see something nice that happens. If I average, then I'll cancel everywhere except right at the boundary of the system.

65
00:05:53,000 --> 00:05:57,410
So you see that there's positive and negative charges. Cancel accepted boundaries over here.

66
00:05:57,830 --> 00:06:05,479
And so that means that traditionally I can rewrite these charges in terms of physical things that I call polarisations and Magnetisation.

67
00:06:05,480 --> 00:06:12,320
So I've got but the density is related to a thing that I call the polarisation, which is just measuring these little dipole moments inside.

68
00:06:12,680 --> 00:06:17,810
And the current is related to magnetisation and these are average, so they only really matter at boundaries.

69
00:06:18,410 --> 00:06:25,430
And so what you can then do is say, well, let's play around with this and put those in the equations, do some rearranging,

70
00:06:25,580 --> 00:06:29,389
and I can introduce new fields which have these sort of unedifying terms like

71
00:06:29,390 --> 00:06:32,959
the displacement field and the rather confusingly magnetic field strength,

72
00:06:32,960 --> 00:06:37,460
as opposed to magnetic field density, terrible historical terminology, but we're stuck with that.

73
00:06:37,880 --> 00:06:44,990
But they call D and H, and the nice thing about these is that these look these equations look like they satisfy the old Maxwell equations.

74
00:06:45,410 --> 00:06:48,890
They have no divergences and they have very simple equations of motion.

75
00:06:48,920 --> 00:06:52,890
So we're very happy because we can work with those and solve them. Okay.

76
00:06:53,880 --> 00:06:58,740
So I want to make another specialisation, which is that if I turned off the fields outside,

77
00:06:58,740 --> 00:07:06,209
if I turned off electric fields and I turned off magnetic fields, the solids go back to being not having polarisation or not having magnetisation.

78
00:07:06,210 --> 00:07:10,710
So I'm not describing you a refrigerator magnet or something that could have a frozen polarisation.

79
00:07:10,710 --> 00:07:15,420
I'm describing things that only are polarised or magnetised when I stick them in a magnetic field.

80
00:07:15,960 --> 00:07:22,590
So there's a simple way to understand that for polar polarisation in particular, if I have a bunch of neutral atoms, the think of a single atom,

81
00:07:22,710 --> 00:07:27,060
the cloud of one atom of the electron is sort of perfectly cancelled when I look

82
00:07:27,060 --> 00:07:30,330
outside by the positive charges because there's a cemetery in the problem.

83
00:07:30,720 --> 00:07:34,680
But if I put on an electric field, I distort that and the system develops a dipole moment.

84
00:07:34,980 --> 00:07:40,050
But since it wasn't there, in the absence of a field, I can see that the direction was set by the electric field.

85
00:07:40,260 --> 00:07:43,650
You can see that that polarisation is just linearly proportional to the field.

86
00:07:44,310 --> 00:07:47,280
And when I do this, these displays and the same thing would be true.

87
00:07:47,550 --> 00:07:51,300
I'm not going to run this argument, but it'll be similar if I did it for Magnetisation.

88
00:07:51,870 --> 00:07:57,270
So what that means is that these coarse grained fields that I introduced are actually linearly proportional to the electric field,

89
00:07:57,450 --> 00:08:01,140
the external electric field, the externally imposed electric and magnetic fields.

90
00:08:01,710 --> 00:08:06,450
Now, of course, this can only be true on average distances, because microscopically that doesn't make any sense.

91
00:08:06,450 --> 00:08:12,600
The real electric and magnetic fields fluctuate, but as long as I'm willing to give up a little bit of information, these things must be true.

92
00:08:13,320 --> 00:08:16,650
Now, it turns out that this can be captured in a very nice way.

93
00:08:17,070 --> 00:08:20,340
I can just go into Maxwell's equipment, make the Lagrangian I wrote down,

94
00:08:20,520 --> 00:08:25,260
and just add two terms epsilon and a dielectric constant in a permeability like so.

95
00:08:25,680 --> 00:08:29,370
But otherwise nothing has changed. What these do will change the speed of light.

96
00:08:29,370 --> 00:08:34,320
So which was one in purest units before, will now become square root of Epsilon Times new.

97
00:08:34,470 --> 00:08:38,430
But that's it. They've changed some properties, but it's all absorbed in those two constants.

98
00:08:39,120 --> 00:08:43,799
So our lesson is that every insulator is effectively a new vacuum for electrodynamics.

99
00:08:43,800 --> 00:08:50,400
That's what insulators are vacuum. So what we want to ask is kind of an insulator provide a vacuum where that

100
00:08:50,400 --> 00:08:55,530
electrodynamics has a hidden missing term that we the data B term that we'd like to get.

101
00:08:56,190 --> 00:09:01,590
So what does that mean for an insulator to do that? So the insulators we normally encounter, as I've said, have this.

102
00:09:02,430 --> 00:09:10,080
And what I want to do is find a way for an insulator to get this additional piece that will check and others wrote down, which is Theta E Derby.

103
00:09:10,710 --> 00:09:16,470
So if I want to ask what that does, may be a good way to do it is to write down how Maxwell's equations change when I put that in.

104
00:09:17,040 --> 00:09:22,200
So there's a bit of painful work to do that. But what I've done is write down the I leave them as an exercise for the reader.

105
00:09:22,210 --> 00:09:27,180
So homework for you, you put that in and what you find is that you get a mess.

106
00:09:27,270 --> 00:09:29,969
It looks not particularly edifying. What if I write it like this?

107
00:09:29,970 --> 00:09:37,200
You just get a bunch of extra pieces in Maxwell's equations, but if I stare at them, I see that actually they can be got off in a very nice way.

108
00:09:38,010 --> 00:09:44,129
This if I pull out a piece here, it just looks like if I change my polarisation to redefine it,

109
00:09:44,130 --> 00:09:49,380
to include this little piece B And redefine my magnetisation, I go back to the old Maxwell equations.

110
00:09:49,830 --> 00:09:53,760
So what I would say is that the stern tells me that a magnetic field,

111
00:09:53,760 --> 00:09:59,370
in addition to introducing a magnetisation to my material, also somehow magically generates a polarisation.

112
00:09:59,820 --> 00:10:06,030
And similarly, an electric field, apart from changing the polarisation of the medium, also introduces the magnetisation.

113
00:10:06,510 --> 00:10:09,959
So what we've got is somehow a material that's responding in a crossed way.

114
00:10:09,960 --> 00:10:12,570
It's sort of E is triggering B and B is triggering.

115
00:10:12,900 --> 00:10:17,940
And maybe that's not so surprising because I've sort of gone off diagonal over here because I have an E metre.

116
00:10:17,940 --> 00:10:25,550
So intuitively it makes sense. And so what I want to ask is how does this effect, which is called a magneto electric polarise ability,

117
00:10:25,790 --> 00:10:30,200
possibly appear in a solid and it's actually not easy to get that in a classical solid.

118
00:10:30,500 --> 00:10:34,400
So to understand where that comes from, we have to go into the quantum theory of solids.

119
00:10:34,910 --> 00:10:38,090
So again, you know, Q groans here for some fraction of you.

120
00:10:39,050 --> 00:10:43,760
So electrons in solids, they describe like electrons everywhere.

121
00:10:43,760 --> 00:10:50,570
In the absence of relativity, apply the Schrodinger equation. But the Schrödinger equation that they satisfy is very special because it's periodic.

122
00:10:50,720 --> 00:10:58,280
If I shift the coordinate by a spacing of in the spacing between atoms, what if everything looks exactly the same?

123
00:10:58,280 --> 00:11:02,660
Because if I just run by one unit cell over the solid looks identical.

124
00:11:03,320 --> 00:11:06,170
And so Felix Bloch, shortly after the advent of quantum mechanics,

125
00:11:06,170 --> 00:11:10,550
Felix Bloch actually pointed out that the eigen states of those equations are very special.

126
00:11:10,760 --> 00:11:15,979
They can always be chosen to be a plane wave and a periodic function that allows us to solve these problems.

127
00:11:15,980 --> 00:11:21,110
So this is the foundation of thinking of the quantum theory of solids. And so I want to take two features of the solution.

128
00:11:21,110 --> 00:11:23,749
So rather than, you know, go to the maths of that,

129
00:11:23,750 --> 00:11:29,120
I just want to give a representative picture of how you might see the solutions of these equations on a solid.

130
00:11:29,780 --> 00:11:35,899
Now we traditionally label the solutions by some number, some label K and index N,

131
00:11:35,900 --> 00:11:41,480
and what I'm drawing here is the different allowed solutions of this problem labelled by the quantum numbers that I have.

132
00:11:42,140 --> 00:11:49,280
And the two things you want to look at from the solution are the first is that if I look at the solutions and I just look at their energies,

133
00:11:49,790 --> 00:11:51,619
there are some energies where I have solutions,

134
00:11:51,620 --> 00:11:58,730
their energies where I don't have solutions, the energies we don't have solutions imaginatively called gaps and insulators correspond to the case.

135
00:11:58,940 --> 00:12:05,780
When you have electrons, you have an electron in each energy level in this region, and then you stop right here.

136
00:12:06,110 --> 00:12:10,420
So to put in the next electron, you have to pay an energy. Set by the Gap.

137
00:12:11,000 --> 00:12:16,479
That's the quantum mechanical statement. The charges are bound in insulators because you have to pay energy to move them around.

138
00:12:16,480 --> 00:12:25,060
They're not mobile. They're bound in these states. And so this is important because if I if I ask about questions of energy is higher than this gap,

139
00:12:25,270 --> 00:12:30,370
then my picture of insulators as effective vacuum will break down because I've got charges moving around.

140
00:12:30,580 --> 00:12:32,569
That's what controls when you say low energy,

141
00:12:32,570 --> 00:12:37,390
you have to say low compared to what it's load compared to the gap the creating expectations that insulator.

142
00:12:37,990 --> 00:12:42,219
The second feature is actually something that we've sort of known,

143
00:12:42,220 --> 00:12:48,490
but its implications were unexplored for about the first 50 or 60 years of the theory of solids, which is the following.

144
00:12:48,910 --> 00:12:52,479
So in it, in free space, we can we have a free particle.

145
00:12:52,480 --> 00:12:55,930
We can label states by momentum because momentum is a good quantum number.

146
00:12:56,410 --> 00:13:00,550
A solid isn't free space, but it's not complete chaos randomness either.

147
00:13:00,670 --> 00:13:08,500
It's got a regular array of sites. And so in that limit, the momentum becomes actually a periodic variable known as the crystal momentum.

148
00:13:08,860 --> 00:13:12,969
In this picture, what that means is that these green dashed lines should be identified.

149
00:13:12,970 --> 00:13:19,660
And really I'm describing things that live on a circle. And so what that means when I think of things that live on a circle are that the

150
00:13:19,660 --> 00:13:24,970
allowed momenta form circles in one dimension or tauri in higher dimensions.

151
00:13:25,150 --> 00:13:31,450
So these are the sort of allowed states. So somehow the states that I have, if I think about their labels, the labels live on a Taurus.

152
00:13:31,540 --> 00:13:33,940
And that's sort of an important fact about solids.

153
00:13:34,450 --> 00:13:38,800
And this the stories is known as the Brill One Zone, and that's where all the action takes place in solids.

154
00:13:39,250 --> 00:13:43,210
So why would I care about labels that live on a Taurus? Well.

155
00:13:44,580 --> 00:13:47,750
That feature has particularly important topological consequences.

156
00:13:47,760 --> 00:13:51,260
So something we're taught in first quantum mechanics course you take, you know,

157
00:13:51,330 --> 00:13:56,969
all introductory courses tell some lies and this is one of those lies that usually doesn't have any consequences.

158
00:13:56,970 --> 00:14:00,870
And the lie is that you don't need you don't ever care about the phase of a wave function.

159
00:14:01,170 --> 00:14:08,940
That's almost true. The one case you care about, the phase of the wave function is if it changes and it changes in a way around a loop,

160
00:14:08,940 --> 00:14:13,799
then you come back to the place where you started. So this is sort of, you know, one of my favourite authors, Terry Pratchett,

161
00:14:13,800 --> 00:14:17,760
has a quote that coming back the way you start it is not the same as never having left.

162
00:14:18,180 --> 00:14:23,040
Well, this is an example of that. So if I wind around this loop and come back, something has changed.

163
00:14:23,340 --> 00:14:28,200
And so that change was actually something that Michael Berry pointed out, known as Berry's phase.

164
00:14:28,620 --> 00:14:33,510
And so people had not appreciated that solids would be such a rich place for Berry's phase,

165
00:14:33,810 --> 00:14:37,980
partly because the wave functions fundamentally have this periodic direction that I can wrap around.

166
00:14:38,310 --> 00:14:42,750
And so if I think of these block states and think about how they move as I change,

167
00:14:42,750 --> 00:14:48,270
these are parameters of these states, these labels, then there are all these phases that they can pick up,

168
00:14:48,270 --> 00:14:55,530
and that non-trivial winding, surprisingly, can lead to new forces on electrons, and those forces mimic electric and magnetic fields.

169
00:14:56,700 --> 00:15:02,190
And in very mathematically identical ways, they can sort of be shown to be exactly equivalent in some ways.

170
00:15:03,060 --> 00:15:06,750
And in particular, such forces can give rise to a theta term.

171
00:15:07,380 --> 00:15:13,860
So I've written a rather unedifying equation over here. So this is something like a vector potential, but it's sort of souped up.

172
00:15:13,860 --> 00:15:20,130
It has additional indices because it's got matrices that every place where used to have a number for the vector potential of electromagnetism.

173
00:15:20,580 --> 00:15:24,270
And it depends on some very complicated way on these microscopic wave functions.

174
00:15:24,750 --> 00:15:29,290
But it turns out that once you've repackage this, this looks like a very familiar form for topologies.

175
00:15:29,310 --> 00:15:33,450
So Jim Simons was a mathematician who discovered these many, many years ago.

176
00:15:33,900 --> 00:15:38,190
He went on to become the chair of the maths department at Stony Brook, and his hobby was investing.

177
00:15:38,190 --> 00:15:42,299
And he left academia to found Renaissance Technologies, which is a sort of hedge fund.

178
00:15:42,300 --> 00:15:44,879
That's one of the more successful ones. Fun.

179
00:15:44,880 --> 00:15:52,320
Funnily enough, in a nice twist, his foundation now funds an enormous amount of research into areas like topological solids, including my own postdoc.

180
00:15:52,330 --> 00:15:56,550
So, you know, very grateful for people who have gone on to do things outside of physics.

181
00:15:56,730 --> 00:16:03,880
Help for us. So coming back to our story, the microscopic details are really quite gory.

182
00:16:03,900 --> 00:16:08,400
So if I looked at one of my textbooks that I look up to these things and I said,

183
00:16:08,400 --> 00:16:13,410
Let's see if I can come up with an elementary solution, it turned out that this particular time is the very last chapter.

184
00:16:13,430 --> 00:16:16,980
It's almost the very last equation of this textbook. So I'm not going to go into that.

185
00:16:17,400 --> 00:16:21,240
So, you know, nobody wants to see how the topological sausage is made. We just want the product.

186
00:16:21,270 --> 00:16:27,360
So what I'm going to try and do is take the spirit of an effective field theorist and just ask.

187
00:16:28,550 --> 00:16:31,620
What are the consequences of solids in the effective field theory?

188
00:16:31,640 --> 00:16:38,840
So let me just keep this term in there, but still think of this thing as a solid state physicists and ask What are the experimental consequences?

189
00:16:38,840 --> 00:16:42,499
And what could I observe in experiments given that this term is sitting around?

190
00:16:42,500 --> 00:16:48,050
And so what I want to do in this rest of the talk is give you illustrations of what the consequences of this are.

191
00:16:48,650 --> 00:16:52,750
Okay. The rules of the game. Clear. Very good.

192
00:16:53,080 --> 00:16:58,190
Maybe it's actually even though we usually wait for questions at the end, it's good to maybe pause here because I want to change gears and go back.

193
00:16:58,210 --> 00:17:04,950
So any questions on this aspect so far? Very good.

194
00:17:04,980 --> 00:17:09,300
Everything was clear. So everyone's going to get first class results on the exam that I'm handing out shortly.

195
00:17:10,620 --> 00:17:14,279
So the first thing I want to do is make some very general remarks.

196
00:17:14,280 --> 00:17:20,070
And again, in the spirit of thinking about theoretical physicists, we know we don't really like to go into details if we can avoid it.

197
00:17:20,340 --> 00:17:26,280
So I want to ask very general things based on two guiding principles, which are topology and symmetry.

198
00:17:26,640 --> 00:17:32,490
Most of you are familiar with symmetry. Both seem that topology is perhaps a new entrant into how we think about physical problems.

199
00:17:33,180 --> 00:17:39,330
So the first thing to remember, and this is something actually conveniently enough, Joe has this on the board over here.

200
00:17:40,520 --> 00:17:44,060
In quantum mechanics. We actually don't care about engines themselves.

201
00:17:44,330 --> 00:17:50,390
We only care about their effect on the equations of motion. And we care about how they enter quantum mechanics.

202
00:17:50,720 --> 00:17:56,450
And really, Lagrangian is like the action term only into quantum mechanics in an exponential.

203
00:17:57,170 --> 00:18:03,200
So that means that if I stuck that in that term, I wrote down always sits in this e to the I time stuff.

204
00:18:03,860 --> 00:18:09,240
So why is that important? That's important because there's a calculation I can't do on the board here.

205
00:18:09,480 --> 00:18:17,190
But if you take my word for it, the structure of electromagnetic fields requires that that integral is actually something that I know.

206
00:18:17,490 --> 00:18:21,960
It's just e to the eye. Peter Times And what data was this number I gave you?

207
00:18:23,280 --> 00:18:27,959
That forces theta and theta plus two pi to always lead to the same physics.

208
00:18:27,960 --> 00:18:31,200
And so if two things lead to the same physics, we just think of them as being indistinguishable.

209
00:18:31,440 --> 00:18:35,879
So we demand that data and theta plus two pi always have the same consequences.

210
00:18:35,880 --> 00:18:41,940
So they must be the same. So theta lives on a circle. So this goes back to Joe's original assertion that axioms are angles.

211
00:18:41,940 --> 00:18:45,930
So this is the proof that it's an angle and not just an arbitrary parameter in a theory.

212
00:18:46,110 --> 00:18:52,940
Because angles live on a circle. The second point is to ask about how symmetry is might constrain this angle.

213
00:18:52,970 --> 00:18:57,379
So notice at this point I have not said anything about what the nature of the media that I'm sitting in.

214
00:18:57,380 --> 00:19:00,680
It's right. I've just told you that it produced this data term. That's it.

215
00:19:01,400 --> 00:19:05,900
Now I'm going to ask about two symmetries that are perfectly reasonable to ask about solids.

216
00:19:06,350 --> 00:19:10,670
The first is the solid has no magnetism sitting around, so it's not magnetic.

217
00:19:10,730 --> 00:19:14,990
There are magnetic solids. I'm not talking about them. I'm just thinking about non-magnetic systems.

218
00:19:15,470 --> 00:19:19,220
The second thing is I'm going to ask that the solid has a centre of inversion.

219
00:19:19,220 --> 00:19:23,750
That's a fancy way of saying the solid has some kind of internal mirror reflection symmetry.

220
00:19:23,750 --> 00:19:29,990
It looks if I send X to minus X, it stays unchanged, so all the atoms will line up exactly where they are.

221
00:19:30,140 --> 00:19:35,810
If I just happen to flip all my axes, so if I use right hand rule or left hand rule, the solid doesn't care.

222
00:19:35,990 --> 00:19:38,990
That's the second step I'm going to do. Now.

223
00:19:38,990 --> 00:19:42,320
Why do I care about these two cemeteries? Well, they do something rather nice.

224
00:19:42,890 --> 00:19:47,410
So. Under the first time reversal symmetry.

225
00:19:47,590 --> 00:19:52,360
What I'm doing is taking space and time and leaving space unchanged and flipping the sign of time.

226
00:19:53,240 --> 00:19:57,590
Electric fields don't care about this, but magnetic fields because they care about currents.

227
00:19:57,710 --> 00:20:01,310
We'll start currents circulate in the opposite direction because you run time the other way.

228
00:20:01,580 --> 00:20:07,729
And so magnetic fields flip sign under this operation. This is why it is very important that I said the solid is non-magnetic because if it

229
00:20:07,730 --> 00:20:12,260
has frozen in magnetic fields then it would not look the same on the time reversal.

230
00:20:12,820 --> 00:20:19,510
And that's one symmetry. The second symmetry is inversion symmetry, which is the exact opposite.

231
00:20:19,750 --> 00:20:26,920
It sends x to minus x, but leaves T unchanged. And since if you think about a solid, it generates a think about an electric field.

232
00:20:27,900 --> 00:20:32,010
It has a spatial coordinate. So if I flip the sign of space, it goes the other way, right?

233
00:20:32,010 --> 00:20:37,379
So. Magnetic fields. It's a bit more subtle way you can see that magnetic fields don't care is you can imagine

234
00:20:37,380 --> 00:20:41,250
taking a current loop and putting it in a mirror and it will still circle it the same way.

235
00:20:41,250 --> 00:20:44,309
On the other side, the magnetic field will point in the same direction and it's mirror image.

236
00:20:44,310 --> 00:20:49,740
So magnetic fields don't care. And so why do what these two terms do, though,

237
00:20:49,860 --> 00:20:57,240
is that notice that they don't constrain ordinary Maxwell electrodynamics at all because Max Electrodynamics has e squared and they have B squared.

238
00:20:57,510 --> 00:21:00,510
E squared doesn't care whether E goes to minus C or dot.

239
00:21:00,720 --> 00:21:01,740
Now that is B squared,

240
00:21:02,190 --> 00:21:10,170
but an E not beta really doesn't like this because it flip sine east is unchanged and b flip sign B stays unchanged in e flip sign.

241
00:21:10,770 --> 00:21:16,680
So if I enforce these symmetries there, if any, if I ask the theory to be invariant under these symmetries,

242
00:21:17,400 --> 00:21:20,700
the equations of the theory have a change in sign.

243
00:21:20,910 --> 00:21:22,290
Every time I see an equation with theta,

244
00:21:22,290 --> 00:21:29,520
I have to put in a minus theta and demanding invariance under these symmetries requires theta and minus theta to be the same.

245
00:21:30,270 --> 00:21:36,209
So this is an elementary thing. If I had if theta were not an angle, it could only be one solution to this equation.

246
00:21:36,210 --> 00:21:41,040
That would be zero. But actually if they doesn't angle this equation as two solutions.

247
00:21:41,900 --> 00:21:45,020
And that's because PI works perfectly well.

248
00:21:45,620 --> 00:21:49,550
You see that if I take pi to minus pi, it looks like I've changed the sign,

249
00:21:49,880 --> 00:21:53,920
but I've already agreed that I'm minus phi and you're just going halfway around the circle.

250
00:21:53,930 --> 00:21:56,720
And it doesn't matter whether you go halfway around this way or halfway around back way.

251
00:21:57,140 --> 00:22:02,120
And so there are exactly two solutions consistent with these Symmetries Zero and PI.

252
00:22:02,630 --> 00:22:08,990
All other values are forbidden. If I have either of these symmetries, I can only have two solutions to these equations.

253
00:22:09,850 --> 00:22:13,360
So a remarkable thing I've understood is that having had the stamps,

254
00:22:13,360 --> 00:22:21,309
I could I claim that the solid has this term fine comes out but if I demand either inversion or time reversal symmetry, I can fix this term.

255
00:22:21,310 --> 00:22:24,850
It doesn't matter any. None of the microscopic details of the solid matter.

256
00:22:25,090 --> 00:22:31,030
This term has to be zero or it has to be five. That's it. So what I.

257
00:22:31,150 --> 00:22:36,879
So that's a nice fact. So for the rest of the talk, I'm going to assume these symmetries and so they're equal.

258
00:22:36,880 --> 00:22:40,600
Zero. I understand what it does because that's just Maxwell and we've gone through that.

259
00:22:40,900 --> 00:22:45,100
So I've got to ask, what happens if I fix theta equals pi? And that's what I want to do.

260
00:22:46,920 --> 00:22:50,580
So what does this mean in a solid what is this non-zero data angle mean?

261
00:22:50,970 --> 00:22:55,620
So we saw that naively when I did the wrote those equations down earlier,

262
00:22:56,010 --> 00:23:01,470
I said I could rearrange them a little bit and see that a magnetic field induces an electric polarisation.

263
00:23:01,500 --> 00:23:05,070
Electric field induces a magnetisation. That's a very appealing picture.

264
00:23:05,370 --> 00:23:12,480
But we do know that what we really care about when we solve things, our equations of motion, we don't care about all of these labels of things.

265
00:23:13,020 --> 00:23:19,040
We care about how the equations of motion change. And so if I do that, I'm actually going to be in for a little bit of a surprise.

266
00:23:19,050 --> 00:23:23,130
So let me there's actually quite a straightforward calculation. So let me take you through it step by step.

267
00:23:24,930 --> 00:23:28,860
These are the equations that I wrote down earlier. These are the modified Maxwell equations.

268
00:23:29,130 --> 00:23:34,750
So you've got this expression of this expression. Right. Remember, there's only two of the Maxwell equations.

269
00:23:34,770 --> 00:23:39,240
We'll come back to that in a second. Two of them were not modified by there being matter in the system.

270
00:23:40,630 --> 00:23:43,180
All I'm going to do now is just take the red bits.

271
00:23:43,510 --> 00:23:48,640
I've got a divergence outside a curve over here, so I have to dust off my vector calculus identities.

272
00:23:48,670 --> 00:23:53,920
Full confession. I used Google to figure this out. I've forgotten half of them, so I have to go back and get the signs right.

273
00:23:54,520 --> 00:24:01,420
I put that in and I rearrange and what I've just done is keep the conventional natural equations in blue on the left and the changes on the right.

274
00:24:01,990 --> 00:24:05,260
And, you know, this is basically a product rule. I have a divergence.

275
00:24:05,260 --> 00:24:08,380
I have two things. Either theta could vary or be could vary.

276
00:24:08,560 --> 00:24:12,410
And so I just have to combine the two changes. Okay.

277
00:24:12,610 --> 00:24:15,740
So, so far, so good. This is just expanding and rearranging.

278
00:24:15,980 --> 00:24:21,140
But now I have to remember that there were two Maxwell equations that have never not entered the game, but they're waiting in the wings.

279
00:24:21,320 --> 00:24:24,870
And those are the two that didn't care that they were sources. They were pristine, unchanged.

280
00:24:24,890 --> 00:24:27,980
They will always be true. So let me see what they do.

281
00:24:28,490 --> 00:24:32,120
What they do is actually cancel two terms over here, arrange them in such a way.

282
00:24:32,120 --> 00:24:35,180
These two equations precisely cancel these two bits here.

283
00:24:35,900 --> 00:24:43,940
And actually, now I have a surprise. If I look at what's left notice of this new term with theta as a constant in both space and time on the right.

284
00:24:44,570 --> 00:24:47,630
Either I have a gradient of theta or I have a time derivative of data.

285
00:24:48,080 --> 00:24:52,639
So what I've realised after doing all this work of getting this term in a solid is

286
00:24:52,640 --> 00:24:57,080
that the equations of motion apparently don't seem to care if this term is there.

287
00:24:57,470 --> 00:24:59,210
They just don't care if it's a constant.

288
00:25:00,240 --> 00:25:05,970
Which is kind of it's a bit deflating because I've done all this work and I've just found out that in the end, nothing seems to change.

289
00:25:07,040 --> 00:25:10,520
But I can engineer a situation where these terms change.

290
00:25:10,700 --> 00:25:16,730
And it's something that's, again, familiar from thinking about the physics of solids is to think about boundaries between media and vacuum.

291
00:25:17,180 --> 00:25:20,260
And at a boundary, the vacuum has theta equals pi.

292
00:25:20,270 --> 00:25:22,370
We know that because there was no either metre.

293
00:25:23,240 --> 00:25:29,780
But imagine that I had a solid with equals by I can ask what happens at the interface between these two regions.

294
00:25:30,210 --> 00:25:32,930
So this is sort of a standard thing that we do a lot of Vietnam.

295
00:25:32,930 --> 00:25:38,930
If you remember your secondary in them, you spend a lot rather in order the amount of time stuck in the rod cam doing boundary value problems.

296
00:25:39,080 --> 00:25:46,160
That's what we're going to do now. So let's look at such an interface that I'm going to talk a little bit about the physics of such an interface.

297
00:25:46,700 --> 00:25:49,849
So I'm going to look at so I've got a region up here with it,

298
00:25:49,850 --> 00:25:56,299
equal SPI a region down here with it equal zero being a so I'm going to imagine this goes on

299
00:25:56,300 --> 00:26:00,110
forever so I don't have to worry about things fringing on the boundaries and stuff like that.

300
00:26:00,800 --> 00:26:04,160
And what I'm going to do is I don't I'm interested in asking.

301
00:26:04,160 --> 00:26:05,660
So normally when I think about a solid,

302
00:26:05,660 --> 00:26:12,710
what I do is put on an electric field perpendicular to the interface and ask how it does polarisation, but that's an ordinary solid.

303
00:26:13,040 --> 00:26:16,279
So what I'm going to do is something that you probably don't normally do in a solid,

304
00:26:16,280 --> 00:26:19,910
which is put on a magnetic field perpendicular to the interface and ask what happens inside.

305
00:26:20,810 --> 00:26:28,280
So I've just translated for the particular coordinate system of the equation we had earlier, so that the divergence of the electric field, remember,

306
00:26:28,280 --> 00:26:31,430
this is just like Gauss's law talks about divergence of electric fields,

307
00:26:31,850 --> 00:26:36,650
and this says that what's on the right side is a source of electric fields, which is a charge.

308
00:26:37,070 --> 00:26:45,230
And so it says that there's a charge that is proportional to the change in theta as I move in the Z direction, but I have an interface.

309
00:26:45,540 --> 00:26:50,690
So that means that theta is constant everywhere but jumps at that boundary just so that boundary jumps.

310
00:26:51,110 --> 00:26:56,030
So what this equation is told me is that there's actually a surface charge density that's triggered.

311
00:26:56,030 --> 00:27:01,640
So I put on a sort of bizarre I put on a magnetic field and suddenly there's a surface charge density that's produced at that boundary.

312
00:27:02,030 --> 00:27:05,629
But this is just a consequence of the forces of acting inside that solid,

313
00:27:05,630 --> 00:27:09,110
that things start moving around so as to produce a charge density on that surface.

314
00:27:09,830 --> 00:27:12,980
Now I force the electric field outside to be zero,

315
00:27:13,220 --> 00:27:18,200
which means that the charge here somehow when it create an electric field somewhere and the logical

316
00:27:18,200 --> 00:27:23,600
consequences that there's an electric field created inside the material parallel to the magnetic field.

317
00:27:24,410 --> 00:27:27,500
So what I've produced is an electric field parallel to B,

318
00:27:27,740 --> 00:27:34,130
and what's remarkable is that that electric field has a strength that's related to that of the magnetic field by the fine structure constant.

319
00:27:34,610 --> 00:27:40,100
So I've done, you know, I've done something remarkable because I've just taken Maxwell's equations with this extra time thrown in.

320
00:27:40,400 --> 00:27:45,290
And I've told you that there is in principle a way to measure fine structure constant by doing an electrodynamics measurement,

321
00:27:45,290 --> 00:27:48,350
which is quite, quite remarkable if you think about where that came from.

322
00:27:49,940 --> 00:27:53,780
I can do the same thing the other way, and that will actually lead us to something quite neat.

323
00:27:54,230 --> 00:27:59,660
So what I'm going to do is now take an interface between three equals zero and theta equals pi.

324
00:28:00,140 --> 00:28:05,720
And let me just set it up before I do that and I'm going to put an electric field now parallel to the interface.

325
00:28:06,500 --> 00:28:10,190
Now, again, if I take this equation, kernels of magnetic fields,

326
00:28:10,190 --> 00:28:14,390
whatever appears on the right hand side should be a current because currents are sources of magnetic fields.

327
00:28:14,750 --> 00:28:24,530
So what it tells me is that there's a current that points in that that set by this that has this gradient and Z times e y.

328
00:28:24,560 --> 00:28:30,530
And if you unpack all of this, you find that there's actually a current that points into the board at this interface.

329
00:28:30,890 --> 00:28:34,370
So it's sort of pointing into the plane perpendicular to the electric field.

330
00:28:35,350 --> 00:28:38,490
And that current looks set up a magnetic field parallel to it.

331
00:28:38,530 --> 00:28:44,109
So just the exact reverse of that effect. And again, this proportionality constant is linked to the fine structure constant.

332
00:28:44,110 --> 00:28:48,220
So again, something remarkable, but actually this effect,

333
00:28:48,520 --> 00:28:53,589
I'll come back to the first effect towards the end, but this effect is actually something familiar.

334
00:28:53,590 --> 00:28:57,190
So on the interface between the theta equals zero and theta equals pi region,

335
00:28:57,700 --> 00:29:02,260
I put on an electric field and the electric field generates a current that's perpendicular to it.

336
00:29:02,860 --> 00:29:05,950
Now, this is actually something that is a very, very, very old effect.

337
00:29:05,950 --> 00:29:09,790
It was discovered by Edwin Hall during a speech this studies in 1879.

338
00:29:10,090 --> 00:29:13,540
This is something known as the hall effect. We don't think of it in the setting.

339
00:29:13,720 --> 00:29:16,510
The hall effect usually emerges when we think about.

340
00:29:17,520 --> 00:29:26,040
An electron could be in some looking at a surface, but I've got a magnetic field out of the surface and I'm trying to push a current through this.

341
00:29:26,610 --> 00:29:30,679
But of course, if I have moving charges, there's a Lorenz force in moving charges.

342
00:29:30,680 --> 00:29:35,560
So the current push a car into one direction, the magnetic field makes it veer away from the direction I want to push it in.

343
00:29:36,120 --> 00:29:40,820
So if I want to maintain that current, I'm going to have to apply additional force to keep it going in a straight line.

344
00:29:40,830 --> 00:29:44,940
Otherwise it keeps going off in the other direction. So it's like having a car with a wobbly sharing wheel.

345
00:29:44,940 --> 00:29:49,130
I have to actually put a bit of force to even keep it going in a straight line if my wheels aren't aligned.

346
00:29:49,350 --> 00:29:51,270
So you can just think of that, that analogy.

347
00:29:51,810 --> 00:29:56,940
And so there would have to be an electric field perpendicular to the current in order to keep a steady current flowing.

348
00:29:57,300 --> 00:29:59,520
So this something that's well known. This is the whole effect.

349
00:29:59,820 --> 00:30:08,729
But what we've heard is that remember that if I go back to the previous slide over here, these coefficients were all fixed.

350
00:30:08,730 --> 00:30:14,910
There's a fine structure constant here. The theta had to jump from zero to pi, so that jump is quantised.

351
00:30:15,180 --> 00:30:18,440
If I have a system with the symmetries, there's no intermediate value of data.

352
00:30:18,450 --> 00:30:23,730
It has to jump from zero to PI because on the surface it can't have any other value, so it has to jump abruptly.

353
00:30:24,150 --> 00:30:28,950
And so what must be true is that that coefficient, there's no freedom in that coefficient.

354
00:30:28,950 --> 00:30:31,590
Alpha was fixed, the jump in theta was fixed.

355
00:30:31,950 --> 00:30:39,410
And so if I translate that into more conventional units, so the whole conduct conductivity is usually called sigma using initial.

356
00:30:39,510 --> 00:30:45,299
So the subscripts x and Y to say that it's a response of a current that's perpendicular to a magnetic field.

357
00:30:45,300 --> 00:30:46,920
So one is an X and the other is in Y.

358
00:30:47,400 --> 00:30:54,240
And it turns out that once you put in all the constants, you get one half a but a minus here because in this particular case, the minus.

359
00:30:54,240 --> 00:31:01,590
But I could rearrange that to get a plus sign in the geometry times squared over H and E square over h is a combination of fundamental

360
00:31:01,590 --> 00:31:07,710
constants that has the units of resistance and in this case the same as resist units of inverse resistance or conductance.

361
00:31:08,280 --> 00:31:13,710
And so that happens to be the same units as conductivity in two dimensions, which is where we are.

362
00:31:14,250 --> 00:31:20,970
So I've got this remarkable result that there's a quantised response that is sort of a universal thing, and there's no freedom in this response.

363
00:31:21,210 --> 00:31:25,650
There's just a half time something. So I've got something that's very, very rigid and fixed.

364
00:31:26,280 --> 00:31:33,690
But in fact, this is really, really, really surprising. The fact that you get this half here is really surprising for a profound reason.

365
00:31:34,350 --> 00:31:40,860
It turns out that if you give me a purely two dimensional system, well, the whole conductance has to be an integer.

366
00:31:40,860 --> 00:31:41,729
Times is quite over.

367
00:31:41,730 --> 00:31:48,120
H Unless you have very, very strong electron electron interactions which we don't have over here, we're imagining the solid is very ordinary.

368
00:31:48,120 --> 00:31:53,220
It doesn't have anything exotic going on. So if you have an integer times it's quite over.

369
00:31:53,460 --> 00:31:59,250
Well, any perturbation I do to the surface I can always think of as something like gluing stuff onto the surface.

370
00:31:59,550 --> 00:32:03,060
Let me glue whatever I can give you a complete freedom to do whatever you like to the surface.

371
00:32:03,060 --> 00:32:08,219
You can glue any two dimensional system you like, but don't modify anything on either side.

372
00:32:08,220 --> 00:32:11,820
You can't put a three dimensional system on there. You can do anything in two dimensions.

373
00:32:12,330 --> 00:32:17,190
Whatever you do, you will actually not be able to change the oddness of this.

374
00:32:17,250 --> 00:32:18,389
So let's see how that works.

375
00:32:18,390 --> 00:32:25,860
I'm proposing doing a sequence of things where I start off with my S quite over to H, so maybe I glue on something which has is quite over.

376
00:32:26,490 --> 00:32:29,670
Sorry, glue on something that has squared over equal.

377
00:32:29,820 --> 00:32:36,690
I get three squared over two h. Let's suppose I glue on something which at minus two is quite a rate that's allowed.

378
00:32:36,900 --> 00:32:41,010
I get minus B squared over two h. So notice I'm changing this number drastically.

379
00:32:41,250 --> 00:32:44,820
But what I can't do is remove that two in the denominator. It's always a half integer.

380
00:32:45,150 --> 00:32:48,900
So this half mess is very strange. It can't be removed on the surface.

381
00:32:49,260 --> 00:32:54,180
And so what we've actually found is something interesting. The surface is actually illegal as a two dimensional system.

382
00:32:54,510 --> 00:33:00,360
No two dimensional system with the same laws of physics that we're put in with weak interactions could ever do this.

383
00:33:00,960 --> 00:33:07,440
So this is okay because the only I defined my surface as one between something that has theta equal zero interact

384
00:33:07,440 --> 00:33:13,800
with spi and such a surface fundamental means the third the third dimension in order to define its existence.

385
00:33:14,130 --> 00:33:18,000
That's okay because it's embedded in three dimensions. So it's a very peculiar property.

386
00:33:20,040 --> 00:33:26,009
And so I should point out that actually understanding I just wrote this down the sigma x y

387
00:33:26,010 --> 00:33:30,270
equals this squared age actually understanding this is one at least three Nobel Prizes to date.

388
00:33:30,270 --> 00:33:35,100
And so, you know, it's it's a deep piece of physics, but it's sort of incidental to the story here.

389
00:33:35,970 --> 00:33:45,270
Okay. So I've done all of this, but I've told you that these consequences of materials which have this data electron action, electrodynamics.

390
00:33:45,510 --> 00:33:50,280
What I haven't convinced you is that there are materials that do this. So you've got to take that on faith so far.

391
00:33:50,670 --> 00:33:57,210
Let me give you some examples of materials that do this, focusing again on the two symmetries I talked about time, reversal, inversion, symmetry.

392
00:33:58,940 --> 00:34:02,030
So there's one little subtlety that comes in.

393
00:34:02,660 --> 00:34:11,210
If I have a solid that has the state equal spy and let's assume that there's never magnetism anywhere, the solid is always time reversal invariant.

394
00:34:12,770 --> 00:34:19,400
Well, it turns out that an insulator can have a hole conductance only if time reversal is broken.

395
00:34:19,490 --> 00:34:23,180
It's forbidden for an insulator that preserves preserves time reversal.

396
00:34:23,810 --> 00:34:27,620
That's a complication, because that seems to contradict what I just wrote down.

397
00:34:28,250 --> 00:34:32,120
Right. So the resolution of this is that somewhere my assumption should break down.

398
00:34:32,450 --> 00:34:38,270
And the assumption that I had to get this piece of physics on the surface was that things were insulating everywhere.

399
00:34:38,740 --> 00:34:41,130
The only way to resolve the tension between these ideas,

400
00:34:41,150 --> 00:34:45,680
the fact that if things were insulating everywhere, the surface would have to have a whole conductance.

401
00:34:45,980 --> 00:34:49,970
But if things were time reversal invariant everywhere, the surface couldn't have a hall conductance.

402
00:34:50,240 --> 00:34:54,410
Something has to give. And what has to give is that the insulating everywhere piece?

403
00:34:54,890 --> 00:34:59,420
So it's actually a remarkable fact that if I demand that the symmetry of nature,

404
00:34:59,420 --> 00:35:03,050
time reversal is present everywhere in the system and the surface and the bulk throughout,

405
00:35:03,560 --> 00:35:08,660
then the surface between equals pi and a theta equal zero insulator has to be a metal.

406
00:35:09,380 --> 00:35:13,520
And the reason that's okay is because metals actually have mobile charges that can get

407
00:35:13,520 --> 00:35:17,000
rid of that surface charge and those surface currents and sort of cancel the effect.

408
00:35:17,790 --> 00:35:24,029
So we've actually discovered I'm taking you through the discovery of actually a profound fact that a time reversal, symmetric system,

409
00:35:24,030 --> 00:35:30,780
which I said equals PHI, is actually in a different state of matter from our vacuum and from an ordinary insulator.

410
00:35:31,050 --> 00:35:33,150
It's something known as a topological insulator,

411
00:35:33,330 --> 00:35:40,170
and it's special because it's interface with our normal vacuum will be a perfect metal as long as time reversal is preserved.

412
00:35:40,680 --> 00:35:46,620
So it turns out that this observation was actually predicting the people predicted this effect comments about 15 years ago.

413
00:35:46,830 --> 00:35:52,979
And one of the people who did this, actually, the person who worked the most individualistic prediction was a graduate student who is working alone.

414
00:35:52,980 --> 00:35:57,750
Rahul Roy, who some years after he made this discovery, spent several years in Oxford as a postdoc.

415
00:35:57,750 --> 00:35:59,850
So sort of a nice Oxford connection there.

416
00:36:00,570 --> 00:36:07,110
So I've just told you about all these special surface effects, but now I've kind of poured cold water on that and said, Oh, the surface is a metal.

417
00:36:07,110 --> 00:36:12,690
So this effect, I told you, is not there. Well, it turns out, though, that that metal itself is very special.

418
00:36:13,350 --> 00:36:20,640
That metal has to remember somehow that if it broke time reversal, symmetry would have to have this half integer conductance.

419
00:36:21,030 --> 00:36:25,800
And it turns out that there's a very special, special consequence of that.

420
00:36:26,310 --> 00:36:31,350
So if you could measure and this is something that experiments next door in the cloud in lab do,

421
00:36:31,800 --> 00:36:37,379
you can measure the surface dispersion of metals very well. And if you measure that, there's a certain thing you can count in another.

422
00:36:37,380 --> 00:36:39,810
If you can see these, it says one, two, three, four and five.

423
00:36:40,140 --> 00:36:45,600
What it's counting is the number of times the dispersion crosses this energy, which is the Fermi energy.

424
00:36:46,020 --> 00:36:52,650
And it turns out that you can you cannot cross it an odd number of times except in this topological special metal.

425
00:36:52,890 --> 00:36:56,010
And so that measurement tells you that into it is what that equals.

426
00:36:56,010 --> 00:37:02,160
Pi exists in this sort of convoluted way, but it's actually evidence that we have an insulator where theta equals pi inside.

427
00:37:02,550 --> 00:37:09,150
Sort of a remarkable fact. But of course, there's another cemetery and that's a bit more amenable to what I want to do today.

428
00:37:09,570 --> 00:37:13,800
So if there April Spy is enforced by inversion symmetry, there's never a problem.

429
00:37:14,130 --> 00:37:18,140
That's because I can't set up the start experiment with inversion symmetry.

430
00:37:18,150 --> 00:37:24,990
So if I gave you a surface that below is one material, above is another material and I send x to minus x,

431
00:37:24,990 --> 00:37:28,470
I've swapped them around because now one material is on top and the other is below.

432
00:37:28,860 --> 00:37:34,769
So a surface can never preserve inversion symmetry. So that whole set of logic breaks down at the surface.

433
00:37:34,770 --> 00:37:39,030
And the surface can always have this property where something can rearrange and be insulating.

434
00:37:39,420 --> 00:37:42,510
And it turns out that other symmetries can do this in various ways.

435
00:37:42,780 --> 00:37:46,919
But for purists, the term axial insulator is reserved for inversion,

436
00:37:46,920 --> 00:37:52,020
symmetric or similar systems where you have data close by in the bulk and it's quantised on the surface.

437
00:37:52,230 --> 00:37:56,129
Break symmetry and it's no longer this perfect metal. So how would you observe that?

438
00:37:56,130 --> 00:38:00,090
Because you no longer have this ability to look to count things on the surface.

439
00:38:00,900 --> 00:38:04,920
So it turns out that these Hoffmann to your whole effects can be done in a clever way.

440
00:38:05,340 --> 00:38:09,540
So there's a effect that is studied in optics, which is called the Faraday effect,

441
00:38:09,750 --> 00:38:13,530
which is that the plane of polarisation, of light rotates in certain solids.

442
00:38:13,890 --> 00:38:17,580
And it turns out a prediction of this material is that there's a quantised Faraday effect.

443
00:38:17,820 --> 00:38:19,620
And so that's something that one can work out.

444
00:38:19,890 --> 00:38:27,180
The only problem with that is that it's quantised if you have a single surface and you can isolate it, but it's very hard to do that with real solids.

445
00:38:27,180 --> 00:38:32,280
Theorists can talk about a single surface, but if you go to a lab, you're going to have to surface this for any finite object.

446
00:38:32,430 --> 00:38:35,250
And, you know, you have to do experiments on finite objects.

447
00:38:35,880 --> 00:38:46,080
So it turns out that this is an ongoing challenge to understand where to actually come confirm that this piece of electrodynamics works out.

448
00:38:46,350 --> 00:38:51,090
So people are still trying to do this in the lab, and they do various ingenious ways of disentangling these effects.

449
00:38:51,940 --> 00:38:56,700
Okay, so to close, I just want to flag one last piece of exotica,

450
00:38:56,700 --> 00:39:03,180
but I think that's sort of interesting because it sort of suggests how solid state materials offer prospects to explore very nice new physics.

451
00:39:03,540 --> 00:39:07,409
So that's going back to this first of the two interface effects I talked about.

452
00:39:07,410 --> 00:39:13,260
So the second one I spent a lot of time on, which is saying that if I put on electric fields, I generate currents.

453
00:39:13,710 --> 00:39:17,040
But the first one was that if I put on magnetic fields, I generate charges.

454
00:39:17,400 --> 00:39:23,430
So imagine I have a sphere with theta equals zero embedded inside a region with that equals by tiny little sphere.

455
00:39:23,940 --> 00:39:28,070
And inside that sphere, I drop in a point source of magnetic field.

456
00:39:28,110 --> 00:39:31,709
The thing that's forbidden by the first by one of Maxwell's equations. But never mind.

457
00:39:31,710 --> 00:39:35,010
For now, let's stick in a source of magnetic field.

458
00:39:35,460 --> 00:39:40,440
If I did that. Well, what I've told you is that, you know, I've got magnetic field everywhere.

459
00:39:40,440 --> 00:39:46,139
I'm going to interface between data equals PI, so actually create electric fields parallel to the magnetic field.

460
00:39:46,140 --> 00:39:49,080
Sorry. Create electric fields everywhere.

461
00:39:49,620 --> 00:39:55,500
If I look outside and I'm only listening to theta equals by region, I don't ask what's going on under the hood.

462
00:39:55,980 --> 00:39:59,070
Then this looks as though I have an electric field of a point charge.

463
00:39:59,610 --> 00:40:01,530
So it looks like if I put in a magnetic monopole,

464
00:40:01,530 --> 00:40:09,390
it triggers an electric field that looks exactly like that of a point charge with a strength related to the magnetic field by Alpha.

465
00:40:10,300 --> 00:40:14,260
So it looks like a magnetic charge triggers an electric charge.

466
00:40:14,770 --> 00:40:18,459
Now, this argument was a bit convoluted, but it turns out that you can shrink.

467
00:40:18,460 --> 00:40:22,870
This data equals zero region all the way to nothing and the argument would still be there.

468
00:40:23,140 --> 00:40:27,219
So monopole moving inside this material would actually carry an electric charge.

469
00:40:27,220 --> 00:40:32,170
And it's something known as the die on. So this is something called the Witten effect, and it's a rather beautiful piece of physics.

470
00:40:32,860 --> 00:40:38,079
So you'd like to be able to observe this, but there's a problem with that, which is that we don't know about magnetic monopoles.

471
00:40:38,080 --> 00:40:38,889
So in a Stanford,

472
00:40:38,890 --> 00:40:44,260
they have an experiment that's designed to search for a magnetic monopole that's just been sitting there looking for a monopole to swing by.

473
00:40:44,620 --> 00:40:49,430
And in 40 years, they've seen one signal that's very controversial. So we don't think that we can find them.

474
00:40:49,450 --> 00:40:51,760
They're not they're not hanging around our universe very easily.

475
00:40:52,240 --> 00:40:58,450
But I think maybe if I've convinced you of one thing, it is that, you know, solid state materials, each of them are in some sense their own universe.

476
00:40:58,450 --> 00:41:02,979
They can generate interesting new back here. And here's an interesting Oxford connection.

477
00:41:02,980 --> 00:41:09,520
So about 15 years ago, a group of people who have either past or very present Oxford Connections predicted that

478
00:41:09,520 --> 00:41:13,600
there are certain magnetic materials that could emulate the physics of magnetic monopoles.

479
00:41:13,900 --> 00:41:19,900
So there's a long and interesting story there. But the neat piece of physics there is that the electric and magnetic fields that are

480
00:41:19,900 --> 00:41:24,490
involved in these monopole problems are sort of emergent electric and magnetic fields.

481
00:41:24,910 --> 00:41:29,020
But very recently people have pointed out that those electric and magnetic fields,

482
00:41:29,020 --> 00:41:35,410
even though they're sort of not the standard ones we think of in max electrodynamics, can actually have their own theta terms.

483
00:41:35,650 --> 00:41:41,229
And so there have been recent papers that predict the this wouldn't effect and having observable

484
00:41:41,230 --> 00:41:46,209
consequences of actin electrodynamics in sort of materials where you can actually do measurements.

485
00:41:46,210 --> 00:41:50,930
So that sort of fictitious sort of special emergent monopoles can have consequences.

486
00:41:50,930 --> 00:41:54,069
And so we can test the sort of rather exotic prediction of field theory in

487
00:41:54,070 --> 00:41:57,430
an experiment that you could realistically imagine sitting on this tabletop.

488
00:41:58,620 --> 00:42:02,160
So let me. I'm sorry.

489
00:42:02,170 --> 00:42:05,670
Going the wrong direction. Let me go back to my sort of two propositions.

490
00:42:05,700 --> 00:42:10,830
The first, I think I hope I've convinced you this is the the the sort of second statement that will check made was that,

491
00:42:11,070 --> 00:42:15,800
you know, it seems clear that we can emulate the physics of axioms in the solid state material.

492
00:42:15,810 --> 00:42:21,390
So just to summarise, you know, we start with the idea that if you think about insulating matter,

493
00:42:21,570 --> 00:42:24,150
then you can view it as a new vacuum for electromagnetism.

494
00:42:24,540 --> 00:42:32,370
And we know now several insulators where this conspiracy of quantum theory allows you to have Axion electrodynamics as the effective description.

495
00:42:32,820 --> 00:42:35,070
And there are many active experimental searches going on.

496
00:42:35,070 --> 00:42:39,450
I gave you a samples of some of them to look for consequences of these Axion electrodynamics.

497
00:42:39,870 --> 00:42:44,640
And I should say that everything I talked about today is treating theta as an angle and not a field.

498
00:42:44,910 --> 00:42:51,450
But there are situations where theta can become a field where this physics comes in, when you have an insulator with some additional complexity,

499
00:42:51,450 --> 00:42:58,409
where there's some magnetism or some other charges moving in very constrained ways and they can generate dynamics for these Axion field.

500
00:42:58,410 --> 00:43:02,610
So you can actually there are also prospects for observing dynamical axioms in the solid state.

501
00:43:04,620 --> 00:43:10,320
So the second point, which is whether or not actions have any physical reality, that study can be useful intellectual exercise.

502
00:43:10,560 --> 00:43:14,220
I have very little to say, but let me close with words of one of my heroes, Duncan Holden.

503
00:43:14,610 --> 00:43:18,890
It's very difficult to know whether something is useful or not, but one can know that it's interesting.

504
00:43:18,900 --> 00:43:20,760
So I hope you'll agree with that and let me thank you.