Ladies and gentlemen, good afternoon. Welcome to the Physics Department and welcome to the 56 Churchill Simon Memorial Lecture.
This lecture is endowed in the names of critics Alexander Lindemann, first discount of Churchill and France,
and Simon Lindemann became the Dr. Leeds Professor of Experimental Philosophy in 1919 and head of the Clarendon Laboratory.
Brunswick and Simon obtains his degrees in Berlin under Nernst and then was called by Lindemann in 1933 to the Clarendon Laboratory,
where he became first reader, and then Dr. Police Professor and head of the Clarendon Laboratory himself in 1956, one month before his death.
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.
As you can see, much of the to of physics in the second part of 20th century is here with many Nobel Prize medallists,
not only in physics, but also in medicine, crick and chemistry.
Kroto. The 50th lecture was Michael Berry, and following him we had Michael Pepper in 2010,
David GROSS 2011, Anthony Salinger 2012, and Robert Loughlin 2013.
Last year we should have had Peter Higgs. Unfortunately, he cancelled too much of much to our regret, the very morning of the lecture.
All these great names testify that physics is full of discoveries.
However, very few such discoveries qualify as paradigm shifts to use the language introduced by Thomas KUHN.
Discoveries such as the laws of electromagnetism. Special relativity.
Quantum mechanics are not only groundbreaking, but completely change the way we view our world as physicists.
One such paradigm shift in the 20th century, perhaps most clearly associated with Soviet physicists left Londo,
was the realisation that changes in symmetry are ubiquitous in describing and producing physical phenomena
as simple as the crystallisation of water in twice and as complex as superconductivity or the Higgs boson.
Another paradigm shift in the 21st century, perhaps, unbeknown to most of the general public, has occurred,
whereby a new concept called topology is taking a role that is not too dissimilar from that taken by symmetry previously.
This paradigm shift is most clearly associated with today's lecture.
It is therefore my great pleasure and honour to introduce to you the 56th Chair, Will Solomon Memorial Lecture.
Professor Charles Keene. Professor Keene is the Class of 1956 endowed term chair, professor of Physics at the University of Pennsylvania.
And I had to read that it was quite complicated.
He completed his bachelor degree in physics at the University of Chicago in 1985 and at the MIT in 1989.
Prior to joining the faculty at Pennsylvania, he was a postdoc at IBM.
T.J. Watson Research Centre. Professor Cahn obtained many awards for his work.
He was the director prize winner in 2012.
Two that together with the shoeshine Jane and Dunkel Haldane for their groundbreaking work on 2D and 3D topological insulators.
He was chosen in the same year for the inaugural class in mathematics and physics for the same assignment investigators.
And he shared the following year the Physics Frontier prizes with Lawrence Malcolm intrusion for their work on topological insulators.
He will speak to us today about topological boundary modes for quantum electronics to classical mechanics.
So please welcome Professor Charles Cahn. Well, thank you very much.
It's really quite an honour to be here.
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,
and also getting the opportunity to experience firsthand some of the wonderful traditions that you have at this historic institution.
So it's really been a pleasure to to come here. And so I'd like to thank Paolo for for introducing my subject,
because the ideas of symmetry and topology are indeed two of the sort of building blocks,
the foundational tools that we have for understanding matter.
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.
And symmetry has been one of the guiding principles in physics.
When you're a student in physics, it's the first thing that you learn that you know,
if you can understand the symmetries of problem, you can simplify it and you can understand it.
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.
Okay, so. So topology is an equally deep and equally powerful idea.
And, and rather than thinking about what, what you can do to a system that keeps things the same,
it addresses the question what stays the same when you change a system?
And so so for instance, if you think about a the surface of an orange is kind of like a sphere.
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,
you know, and you could do that continuously and nothing changes.
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.
OC Even though there's no symmetry, you know, between them, but they're different from a doughnut.
To get to a doughnut, you'd have to poke a hole in the atmosphere, and that's against the rules.
Okay, but a doughnut is kind of like the surface of a coffee cup, okay?
Because you can stretch it and mould it, and the whole of the doughnut turns into the handle of the coffee cup.
Okay. And so this idea of topology asking what what are the sort of fundamental things that stay the same when you change
something is an is is a an additional powerful tool that we have for now understanding the phases of matter.
And so this is a subject which over the past ten or 15 years has really developed a lot.
And really, it's it's it's our, you know, so of course, symmetry and topology are old ideas.
They've they've been present in physics for for for many, many years.
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.
And so what I'd like to do today is to describe to you some various situations where where this is crucially important.
And in particular, I'd like to describe the sort of new quantum electronic phases of matter that we have discovered.
And, and explain to you how symmetry and topology sort of intertwine to to to give us an understanding of those materials.
And after I do that, though, this is going to be a little bit of a schizophrenic talk,
because I'm going to shift gears and talk about a completely different set of physics problems,
not not quantum classical physics problems, where remarkably,
the same kind of mathematical ideas that we use to understand these materials show
up in these classical mechanical modes or what I will call ISO static lattices.
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
they can be understood with similar using the similar sort of basic principles of physics that that of symmetry and topology.
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,
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.
The most recent work was a collaboration with another colleague of mine at the University of Pennsylvania, Tom Levinsky.
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.
And and I want to start with the most basic phase of matter.
Okay. And and the most basic phase of matter is the the insulating state.
Okay. Now, you know, you might think the insulating state is the most boring state.
It's basically the state where the electrons do nothing.
Okay. But it's something that we tell our students about.
You know, in in beginning physics, what we say is that in an insulator.
It cannot conduct electricity because the electrons are bound to the atoms and they're stuck to the atoms.
And so. So every electron is sort of stuck. It can't move.
Electricity cannot flow. Now, actually, this fact that the electrons are stuck to the atoms is a uniquely quantum mechanical fact.
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,
quantum mechanics tells you that the energies that electrons can have are quantised.
They can only have specific energies. And so what happens in an atom, in a in an inert atom, say, like like argon,
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.
You can put an electron it. Okay.
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.
Okay. And so for that reason,
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.
Now, of course, a crystal of, you know, inert argon atoms is not a very interesting material,
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.
And in that case, one has a more interesting situation where the different atoms interact with one another.
And so so the the theory that describes this was really one of the crowning triumphs of quantum mechanics in the 20th century.
And so this theory for understanding the electronic structure of crystalline materials is called the band theory of solids.
Okay. And and this is really probably one of the most consequential theories that of matter that
we have is the theory that's responsible for our understanding of semiconductor materials,
for example. It's the theory that underlies this laser pointer.
It underlies the cell phone in your pocket. All of the information technology that that that that we are experiencing today ultimately results from
this from the deep understanding of of materials that we have based on the band theory of solids.
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.
And what happens when you make them in a crystal is they broaden out and form a band of energies.
And in an insulator, the the occupied energy states are separated by an energy gap from the empty energy state.
So an energy gap separates the so-called conduction band from the valence band.
And so so due to the existence of this energy gap, this insulator is electrically inert.
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,
in this valence band that's enabled enables, you know, lots of technological advances.
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,
in the sense of topology that I introduced before,
which you can imagine sort of smoothly deforming this one into this one, making the atoms get a little bit further apart, further apart.
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.
Okay. So there's a sense in which the the insulating state of a say,
intrinsic semiconductor is kind of the same as a the insulating state of of of a, you know, gas of inert atoms.
Actually, it's also kind of the same as the vacuum.
Okay. Now, you know, the vacuum actually is not as empty as you might first think.
So if you think in terms of the relativistic theory of quantum mechanics, Dirac's theory of quantum mechanics,
the vacuum has a valence band to okay the valence band for the for the vacuum is if it is the positron.
So if I remove an electron from this valence band that's what the antiparticle of the electron is the positron.
And so the only difference is that the energy gap here is a million times bigger.
Okay? So there's a sense in which all three of these are the same.
Okay? And, and that sameness is where the idea of topology enters the discussion of the band theory of solids.
Okay. And what's remarkable is. That these ideas of topology and band theory were left undiscovered for so long.
It was a treasure that was sort of waiting to be uncovered. So.
So. So the question you can ask.
So. So. I will say that insulators are topologically equivalent if they can be smoothly deformed into one another without closing the energy gap.
Okay. So that poses the question.
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?
Is there an analogue of the doughnut? Okay. And the remarkable answer is that, yes, there is.
And these electronic phases are something that are very interesting to study.
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,
that you always try to find the simplest example of something first.
And the simplest example of this actually has a long history.
The so the simplest example is what you can have if you have a, a system which is one dimensional.
So it's defined on a line. And so the system I have in mind is a conducting polymer.
So this is a chain of atoms, it's a chain of card carbon and hydrogen atoms called poly acetylene.
And poly suddenly is an important material.
It's sort of the first the simplest example of a conducting polymer now.
So so if you just have a chain of these carbon atoms, each one has has an electron on it.
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.
And so this means that there is no energy gap separating the occupied electronic states from the empty electronic state.
So electrons can sort of move around with little cost and energy.
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.
What Pauli acetylene would rather do is it would rather die,
arise and have the pairs of atoms move a little bit closer to each other and and a little bit further apart.
So, so the so it forms dimers where the bonds are a little bit stronger.
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.
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,
having all the occupied states go down. But the interesting thing is that now it's an insulator.
Now there's an energy gap which which separates the occupied states from the empty states.
Okay. So but but that's the thing that's nontrivial about this is that there are two ways you can do it.
You can either demonise going short, long, short, long.
Okay. And that would be if I move the green atoms to the left.
Okay. Or I could die. My eyes going long, short, long, short.
Moving the green atoms to the right. Okay, those are both insulators.
But what I want to argue to you is that in this sense that I defined,
they are topologically distinct insulators in the following sense that that if I tried to smoothly go from this situation to this situation,
I'd have to go through here where the energy gap is zero. Okay.
And so the A phase and the B phase are separated.
If I try to go if I try to tune myself from one to the other,
I have to go through a quantum phase transition, a critical point where the energy gap goes to zero.
Okay. I mean, because and so that's exactly what I mean by topological.
Like, you know, all the Fe all the states over here are topologically equivalent.
All the states over here are topologically equivalent. But these and these are different.
Okay. And so so something one can do.
And I don't want to sort of delve into the mathematics of how one describes this,
but by understanding the structure of the electronic states in the valence band, one can define what's called a topological invariant.
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.
So so if you you can imagine if you have a rubber band wrapped around a pole, okay,
you could distinguish this from you can distinguish whether it's wrapped around or not.
In fact, it could wrap around any integer number of times.
And the distinction between the A and the B phase really is sort of like this this this winding number.
Okay. I will say that this this there's there's a there's a cow.
I always have a little star here. So there's always a little bit of a caveat, a technical caveat with this.
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.
So symmetry is actually playing a role here. And that symmetry is actually an important one which is going to show up again,
which is a symmetry that relates to the conduction band of the Valence band.
So in the simplest picture of polys settling the conduction band and the Valence band are mirror images of each other.
So everything going on at a positive energy is also happening at a negative energy.
So, so battery case. So, so the two phases are topologically the same.
So you could ask yourself, okay, what, what consequence does that have?
And the most important consequence of this topological distinction is what happens
on the boundary where the two distinct topological phases meet each other.
Okay? And so you can imagine, let's suppose you have a phase on the left in the B phase on the right.
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.
Then you have an atom that's left over in the middle here, and it's not talking to anybody.
And so you have a big energy gap from from from from on both sides.
But then there's this one state that's left over in the middle here that's at zero.
Okay. Now you can ask yourself, okay, what happens if I turn these bonds up?
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.
And that means that this state of zero can't go anywhere.
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.
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.
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,
and it's something which has been discovered and rediscovered many, many times over the years.
So I think the first discovery of this idea of the topological boundary mode was by the field theorists,
in particular Roman Jacques-yves, and he was interested in certain sort of one dimensional field theory problems.
And then some time later, Bob SCHIEFFER and his collaborators were interested in poly acetylene and were interested in particular these these domains,
these domain wall states in polish acetylene.
And they sort of rediscovered this this idea that you have this topological zero map.
Now, in the years since, what we have learned is that there are many, many more examples of this phenomena.
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,
at the boundary between two topologically distinct insulating phases, there exist some topologically protected low energy states.
Okay? And so these top these, these, these low energy states are states that you can't get rid of.
Okay. So. So there are many, many examples of this.
Probably the most famous one is what is called the Quantum Hall effect.
And so the Quantum Hall effect is something again, it happens to electrons.
It happens to electrons when you combined confine them to a two dimensional plane.
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.
Now, what a magnetic field does to electrons is it makes them want to go around in circles.
So that's what electrons do in a magnetic field.
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.
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,
it does kind of the same thing that it did in an atom. It makes the energy levels quantised.
And so these quantised energy levels in a magnetic field are called landau levels.
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.
And so so this is kind of like an insulator, right?
You have an energy gap that separates the occupied electronic states from the empty electronic states.
Okay, so it's just like an insulator, but it's not an insulator.
Okay. It turns out if you apply an electric field, if you try to put a put a voltage on on on this state,
then what the electrons who are going around in a circle, if you apply an electric field,
the electrons start moving perpendicular to the electric field. That's what happens if you have motion in a cross E and B field.
And so so actually this defines a.
So you say you put a field this way. You've got a current this way.
This defines what's called the hall effect. You get a current perpendicular to the field.
And in fact, what's remarkable is if you measure this current, if you measure the hall current, the current going perpendicular,
it defines a conductivity, if you will, which is incredibly accurately quantised in units of this fundamental unit of conductance.
So this is a combination of fundamental constants, you know, the electric charge and flex constant.
This defines a unit for the resistance or the or the conductance.
And what is measured is that you measure these incredibly sharp plateaus in the hall conductance,
where this integer here is something that you can measure to one part in a billion.
It's an incredibly accurate quantisation. And and so so what's the origin of this?
Well, the origin of it is really topology. What you are measuring here is actually a topological invariant.
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,
David Thalis and his collaborators.
And he pointed out that the hall conductivity is, in fact a topological invariant that has a number in the mathematics,
in mathematics called the chern number that characterises a two dimensional insulator, basically.
Okay. And so, so, so. So you get this. So, so, so.
So this quantised hall conductivity really is a topological invariant.
Now, as I said, this topological class classification has consequences on the boundary where you have one topological classmates another.
So if you have an integer, a quantum hall state here and the vacuum which is an equal zero,
then what happens on the on the edge on the boundary is actually a funny, familiar phenomenon.
In the Quantum Hall effect, you get a what's called an edge state.
And so if you think about the motion of the electrons, the electrons are going around in circles in the magnetic field.
Well, if you put a wall here, then the electrons can bounce off the wall and then they can skip along.
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?
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.
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.
And in quantum theory, we give them a fancy or name, we call them one dimensional chiral, Dirac, fermions, and and these one dimension,
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,
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.
And this perfect transmission of these chiral edge states is really underlies
the fact that you can measure this hall conductivity to one part in a billion.
So there's there's something very deep about the topological protection of these chiral edge states.
Okay. So now for many years it was believed that this was kind of the only non-trivial topological states that you could have.
And it's a little bit of a contrived situation because because, you know, these edge states, they go one way.
If you were to take a motion picture of this and run it backwards, then the edge states would be going the other direction.
Right. If you ran it, ran the motion picture backwards.
And so what this means is that this this this this motion of the electrons violates the symmetry under the reversal of time.
So time reversal symmetry basically ask you whether whether if you run the motion picture backwards, whether things look the same or not.
Now, the laws of physics. Are invariant under time reversal symmetry.
If you if you had ran your motion picture backwards, what you saw would obey the laws of physics.
Okay. But here,
what the reason time reversal symmetry is violated is because you have to apply a magnetic field and a magnetic field violates time reversal,
symmetry. And so. So the question one could ask is, what if you don't have a magnetic field?
What if you do have symmetry under the reversal of time?
Could there be anything interesting left? Okay, so this chern number dn would have to be equal to zero.
Okay. And so for many years it was believed there was nothing left.
But in the last ten years we've discovered that in fact there is something interesting that is left.
So, in fact, if you have time reversal symmetry, the symmetry under running the motion picture backwards,
then in fact there are actually two distinct classes of insulators.
There is a trivial insulator, which is like the vacuum, but there is also a topological insulator.
And unlike the Quantum Hall State, where,
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.
And and the the interesting thing is what happens on the boundary between the trivial and the nontrivial and on the boundary what,
what has conducting edge states which are sort of like two copies of the edge states that you have in the Quantum Hall.
Fact, you have one that's going one way and one that's going the other way.
So so if I, if I, if I run it backwards in time, then, you know, it still looks the same.
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.
We actually realised theoretically that this state of matter could exist before we had seen it in the real world.
And so there are various, you know, so once we knew it was possible, then the question is how can you make it?
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.
These are these are sort of well known semiconductor materials.
And and to cut a long story short, shortly after the prediction that that this phase was possible,
experiments were done on these structures which which confirmed that that this structure did, in fact, have these these special kinds of edge states.
Okay. So this is what happens in a two dimensional insulator.
There's also a three dimensional topological insulator.
So in three dimensions, again, there's there are two classes of insulators and in the non-trivial insulator,
again, it's an insulator on the interior, but on the surface it is a very special kind of conductor.
Okay. And it's and it's a special kind of conductor where the conducting state, you can't get rid of it.
Okay.
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.
Okay. And again, the, the the theoretical predictions of these of this state came first.
And then after the predictions and the predictions of real materials came experiments.
In this case, this is an experiment called angle resolved photo emission spectroscopy,
which is a way of measuring the electronic structure of the surface of a material.
And what one observe here observed here is sort of almost a textbook perfect version of what's called a surface Dirac cone,
which is the prediction for what the electronic structure of the surface is.
And so so the discovery of these materials has sort of opened a floodgate of, of, of activity in both experimental and theoretical physics.
And so this is, you know, so these materials are very easy to to study.
And there's there's really a lot going on. So so what?
So I want to give one more example of a topological boundary mode that can occur in quantum electronic systems.
And, and so this is one final example, which is what is called topological superconductivity now.
Now superconductivity. This is a rather advanced topic in physics.
I'm not going to I'm not going to be able to explain to you in detail what superconductivity is.
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.
A superconductor is really just like an insulator. Okay?
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.
Okay. But the price that you pay for having this simple picture.
Is that is that the conduction band and the Valence band actually really are mirror images of each other.
Okay. You remember when I introduced this idea of the mirror, the particle hole symmetry before?
It was sort of an approximate symmetry here. It's an intrinsic symmetry which relates the conduction band to the valence band.
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,
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.
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.
So what? So the interesting question then is, is could these things, you know, you know, sort of band structures,
if you will, that have this intrinsic particle hall symmetry, could they have topological classes?
And the way you can think about that is you could ask, well, what happens at the end?
And at the end of a one dimensional, topological superconductor, it's possible to have a state that's at exactly zero energy.
Okay. And if you have this state at zero energy, it's topologically protected because if it tried to move away from zero energy,
then its partner would have to appear out of nowhere. And that can't happen.
Okay. So. So this zero mode is topologically protected.
Okay. And so. So. So you could ask, well,
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?
Okay. Now, this zero mode is actually a a very magical kind of thing,
and it's something that we very, very much would like to make happen in the real world.
Okay. Now what? So. So remember I told you that we have this redundancy?
Adding a particle at energy is the same as removing a particle of energy minus E.
So that means that adding a particle at zero energy is the same thing as removing a particle at zero energy.
So that means that this particle is its own antiparticle.
Okay, I put one in. I put another one, and poof, they're gone. Okay.
So. So this is a particle which is its own anti particle.
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.
So my R.A. was a I was a brilliant theoretical physicist, a protege of Enrico Fermi's,
who discovered shortly after Dirac invented relativistic quantum mechanics,
that that a consistent interpretation of Dirac's equation was that it's possible to have a particle that is its own antiparticle.
Okay. Now, Maiorana had a tragically short career, and shortly after shortly after that discovery, he disappeared.
And it's a big question what happened to him. Okay.
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.
Of course we don't really have the fundamental majorana fermions is micron.
So these majorana fermions are sort of emergent, if you will, maiorana fermions.
But they're something we very much would like to make.
And the reason we would like to make it is due to an idea from a modern day genius named Aleksey Katsav,
who realised that if you could, if you could make these,
these majorana bound states at the end of a topological superconductor,
then they can give you a new method for storing and manipulating quantum information.
And and so by making these, they could give you a route to making a kind of quantum computer.
Okay. And now this is something we are not close to implementing yet.
We're not ready to make a quantum computer out of these things.
But this gives tremendous motivation for us to make this happen in the real world.
Okay. And so so there's a huge effort going on in the world to try to discover these a mirror on a Fermi.
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.
I just want to say that there are various proposals for how one can implement it to realise these majorana zero modes.
So these are, these are the majorana zero modes that one can hopefully make.
And, and there is very encouraging evidence that these majorana zero modes may have already been discovered.
Okay. So the challenge is to make sure that these experiments are really right.
And there's there's there's huge debates going on as we speak about these things.
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.
Okay. So what I'd like to do in the remaining time is to shift gears completely.
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,
and it shows up in sort of many different contexts in this electronic quantum electronic states of matter.
And so what I want to show you now is how this very similar idea shows up in a completely different situation.
Okay. And so the set of problems that I'm going to describe, you are actually even older than the problems of quantum mechanics.
They predate quantum mechanics itself. So these are a set of problems that go back to the days of of Maxwell.
Okay. So so now, of course, many of you will have heard of Maxwell as the Maxwell who created the the laws of electromagnetism.
But Maxwell also had other interesting interests as well.
And one of them was the interest of sort of what you might call structural engineering.
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?
So so imagine you're trying to build a house and you build a house out of, you know, bars and springs like this.
You know, if you build it like this, you might worry that something like that would happen.
Okay. And then your house would fall down. Okay. So.
So what's wrong here is that there's a way of there's a there's a degree of motion
that doesn't have any restoring force that doesn't that that that that is floppy,
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,
say, then how many of these floppy modes are there?
And so the insight he had was that, well,
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.
So in two dimensions each, each, each mass can move in two directions.
So each mass has two degrees of freedom. Okay.
And then you need to compare that with the number of constraints that you have, which is like the number of springs.
And so what Maxwell showed is that taking the difference between the number of degrees of
freedom and the number of constraints gives you a lower bound for the number of floppy amounts.
So if you don't have enough springs, then, then, then then your house will fall down.
Okay. Now, of course. So what you might do, I, I, you know, if you wanted to make.
A better house is you'd probably put some more springs in and sort of stiffen it up.
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.
So Maxwell's rule is not an equality, okay, because I could have lots and lots of springs, lots and lots of bonds.
So Maxwell's rule is an inequality. Okay, now, so this is actually a problem that is of interest in the Structural Engineering Committee.
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?
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,
if I, if I tightened the Red Springs. So the Red Springs are under tension.
They're pulling in and I loosened the Green Springs so that they're under compression.
They're pushing out. Then the forces on all of the masses is equal to zero.
They can those forces can cancel each other. And so this is an equilibrium situation.
There are no forces. So this is what is called a state of self stress.
Okay. And what. And so Khalidi is it was an engineer who in the 1970s revisited this problem.
And what he was able to show is that Maxwell's counting rule can be turned into an equality.
If you take into account the state of self stress rates of self stress.
So the difference between the number of floppy modes and the number of these,
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.
Okay. So this is this is the Maxwell problem. And so so so what?
So where I entered this, thinking about this.
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.
And he's my colleague at the University of Pennsylvania. And he's been interested in these problems.
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.
So these would be situations where your system is just on the verge of mechanical instability.
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.
So ISO static means that that the number of of of sites and bonds are such that this count is equal to zero.
Okay. And so, for example, a square lattice is like that.
Each site has two degrees of freedom and then there are sort of twice as many bonds as there are sites.
This lattice is called a kagome, a lattice, and that also has the same property in three dimensions.
There's a lattice called up hydrochloride lattice that has this property.
Okay. And so, so so these are this is so so the interesting question is what happens to these lattices?
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
are of interest to soft matter physicists and statistical physicists and also engineers,
problems like rigidity, percolation jamming, network glasses.
And so these are these are the set of sort of physics problems that these models can be applied to.
Okay. And so so when Tom thought about this, he realised that there's an interesting situation.
And the interesting situation is to think about this, this category, a lattice.
And so here's what Tom showed. So, so if you if you if you just have one of these cockamamie lattices.
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.
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,
you know, figure out what what the motion of this kind of thing would be.
Okay. So so one thing you can see is if you have a lattice like this, it has it has lots of floppy moats.
Okay. So they, you know, the things can sort of go like this and there's no restoring force for that.
Okay. Now, there are also lots of states of self stress.
So so if I have a straight line of bonds like this, then, you know,
if I put all of these bonds under tension, then then all the forces are going to cancel out.
Okay. And so, so that makes sense. You have lots of floppy modes, lots of states have self stress.
So this count, you know, so it's perfectly static.
So that means the number of these has to be equal to the number of these. So that's fine.
Okay. There are lots of both of them.
But what Tom realised is that he can do a very simple modification of this kagome lattice by twisting the triangle.
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.
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,
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.
So that's not a state of self stress anymore.
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.
Okay. And so this so the system sort of stiffens up. Okay.
But the thing that's remarkable about this,
so this this is what happens if you imagine sort of wrapping it periodically around so that there's no boundary.
What Tom realised is that when there is a boundary, something very interesting happens, okay?
And in particular, so again, this is this is a problem that one could imagine for a second year physics student,
you know, solving for the motion of these masses and springs.
And what Tom discovered is that when he cuts open this lattice so that there's a free boundary,
then even though we got rid of all the zero modes in the bulk, there are floppy modes localised on the boundary.
Okay. And so. So the way a physicist would analyse this problem is by looking at the sort of natural
frequencies of vibration of this system as a function of the sort of wavelength in the,
in the X direction. And. And so what one, so want to have a picture like this of the normal nodes.
And the important thing here is that there are all of these zero frequency modes.
Okay. And now. So. So actually, before Tom did this calculation, he knew that these zero modes had to be there.
Okay. He had an ingenious argument based on this Maxwell accounting rule that that that guarantees that these zero modes have to be there.
But once you look at these, you know, I mean, so so you can change things.
You can move the triangles a little bit. And these zero modes don't go away.
They're stuck there. It's like their topological.
Okay. So now so Tom, he, you know, his office is down the hall from mine.
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.
And so we bump into each other in the hallway and we talk and and and so so one day,
a couple of years ago, you know, Tom came up to me and he said, you know, you know.
Are my boundary moats related to your boundary moats? And so I said, no, Tom, no.
You know, I you know, I'm I'm doing, you know, topological quantum, you know, states of matter.
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.
But Tom, you know, he's a he actually understood some things that I didn't understand and he was persistent.
And so he came back. He kept coming back. Came back to me again.
So finally I agreed to sit down with him and talk.
Talk about it in his office for a while, tend to think more seriously about it.
And so what I decided to do, I started to come up with the argument that would put this to rest.
Okay. And so here is the argument that I constructed. I said, Tom, look, I'm doing quantum mechanics, okay?
So quantum mechanics is based on the Schrödinger equation, which is a first order differential equation in time.
And what one is interested in solving the Schrödinger equation is one is interested in solving for the spectrum of the Hamiltonian,
which are what are called the eigenvalues of this Hamiltonian matrix.
And the important thing for me is that the eigen, the Hamiltonian has positive and negative energy eigenvalues,
which form the conduction band and the valence band.
Okay. And all of my topological band theory is all about Topologically classifying the valence band.
So that's what I'm about. Okay. But Tom.
Tom is just solving Newton's laws. F equals M-A and Newton's laws are second order equations in time.
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,
you have a, you have a a matrix, two diagonals.
You want to find the spectrum of of this what's called dynamical matrix.
But this dynamical matrix is very different than the Hamiltonian in the sense that it's
eigenvalues are all positive and then the frequency squared is equal to the eigenvalues of this,
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.
The frequency is always positive. So I said, Tom, you know, these problems are completely different.
But then. When I said it that way. I was reminded of a story.
And. This is the story of the legendary Solvay Conference of 1927.
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.
But for me at least there there are a number of faces that I didn't recognise.
But what was remarkable is that when I went and looked them up,
I realised that I knew all of their names because their names sort of live on in the nomenklatura of of our field.
And so this is this is actually a remarkable thing. So, so in that case,
this Solvay conference was a conference that brought together the pioneers of quantum theory to debate that sort of new emerging subject.
So in particular, Niels Bohr, you know, the famous grandfather of quantum theory,
was there, and also Paul Dirac, who was young and not yet quite so famous.
Okay. So the story goes that that Bohr bumps into Dirac during a coffee break and asked him what he's up to.
And so Dirac was a man of notoriously few words, and he said, I'm trying to take the square root of something.
So Dirac. Trying to take the square root.
Niels Bohr was okay. You work on that?
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,
because Dirac was trying to take the square root of the Kleine Gordin equation to unify quantum mechanics with the theory of relativity.
And Dirac came up with a ingenious way.
So the question is, how do you take the square root of this without taking a square root?
And Dirac came up with an ingenious way to do it by turning it again into a matrix and multiplying the two matrices together.
And in doing so he discovered the Dirac equation.
Now the direct equation, in addition to explaining the microscopic origin of the spin of the electron.
You made another prediction, which is that the electrons have a valence band.
Which is the which gives us the antiparticle of the electron.
Okay. So, so, so, so. So Dirac discovered the positron.
But the positron is really just the valence band. Okay.
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.
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.
Now I have to apologise. This is a little bit of a technical discussion here.
And I promise you, this is the only technical discussion.
But but, but it's so cool that I just for the physicists in the audience, I want to take you through this briefly.
Okay. So so the idea here is the following.
So the question is, so this dynamical matrix has a very special form because, you know, really.
So the what I want you to think about is think about what the energy of this system of masses and springs are.
So the, the energy of a spring is just one half k x squared is just hooks, hooks law for the springs.
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.
So each spring, the amount I stretch each spring by depends on the extension of the springs by a q.
And what that means is that the energy is q. Q transpose.
Okay. So it's almost. So the dynamical matrix is almost a perfect square.
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.
This has exactly the same structure as supersymmetric quantum mechanics.
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.
Okay. And to cut a long story short.
One can combine these two matrices into these these two operators into a bigger matrix, sort of like what Dirac did.
Okay. And then this thing is a perfect square.
Okay. And so now the thing that I so now I have taken the square root of Tom.
And now I have a Hamiltonian that I can interpret as a quantum Hamiltonian.
Okay. And so once I have this, then I can apply everything that I know about topological band theory.
So this one has a valence band. Okay. And.
And so. So just if there are any aficionados in the audience, the thing you have to do is understand the symmetries.
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.
So this is something that I completely recognised anew. Once I knew this, I knew how to proceed.
Okay. And so by doing this,
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,
you can have floppy modes if you have an interface between two topologically distinct regions.
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.
But now I have localised floppy modes and then there are also localised states of self stress on this boundary.
Okay. So now just in the last few minutes, I want to I'm not going to go through this technical stuff,
but I want to show you this just because it's kind of fun.
Okay. And so so, you know, once I had this mapping between the sort of classical mechanics problem and the quantum electronics problem,
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?
Okay. What's the simplest example of a topological phenomenon?
And so the simplest example in topological band theory was, I already told you it's the sushi for here tomorrow.
Is this one dimensional, one dimensional polymer. Okay.
And so, so so this poses the question, what's the mechanical analogue of that?
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.
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.
It's a rigid bar. And then these blue masses can rotate around pivots.
Okay. And then they're connected together by springs. Okay.
So in this case, you can imagine that you could push this and everything would all the blue things would move to the right.
And that would that would be sort of a floppy motion.
This is the analogue of the metallic state of the sushi for Jaeger model.
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.
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.
Okay. This guy can't move without stretching a spring either.
Okay, so these so these are the mechanical modes.
All have an energy gap for for for these motions.
But look at this guy. At the end, he can move. Okay.
That's your zero. Now, of course, you might say, well, this is an artefact of them being parallel.
If you if you do it like this, then actually what you find is there's still a zero remote.
If you have an interface now between one going to the left and one going to the right, there's a zero mode there.
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,
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?
And but I never really, you know, I don't know, had the energy to sit down and do it.
But. But my friend Vincenzo Vitale, who's a professor at the University of Leiden, he got very excited about this.
And so he got so excited that he convinced his boss to to give him a laboratory.
He's a theorist, but but he convinced his boss to give him a laboratory,
which he has stocked full of Lego to be able to put together and make all these kinds of mechanical things.
And so I went and visited him in Leiden, and he very generously gave me one of his toys.
And so so let's see if I can do this here.
All right. So this is the.
Is the. Is what I have here.
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.
So it's like the springs are infinitely stiff. Okay.
So what you can see is that indeed everything is sort of, you know, locked here.
It can't can't move without doing it, but.
There it is. So. So this is like a very low tech micron to zero mode.
Okay. And so so so this really is a topological boundary mode.
Okay. And, you know, it can exist.
You know, so so now I have a situation. I moved it. So I have things.
Everything is rigid here and rigid here. But I have a boundary mode in between them.
Okay. You would never discover this boundary mode by counting sight.
You know, you never discover where it is by counting locally the number of degrees of freedom in the number of constraints.
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.
And and actually, I kind of regret it, because when Vincenzo made this, he actually discovered something that we didn't realise, okay.
Which is that, you know, so one question you could ask is, well, what happens if you go beyond the linear regime?
You know, what happens if you keep pushing the zero mode? Well, if you keep pushing this zero mode, then.
It goes all the way down to the other side. So this thing, it's almost like a particle, right?
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.
And so so this is something that, you know, is a subject for for for for more investigation.
Okay. So I think what I want to do is I want to finish.
And so the main lesson that I wanted to get across here was that this phenomenon of of topology and topological
boundary modes is this really is sort of an elegant mathematical construction and it has applications in,
in, in very diverse venues, the topological electronic phenomena as well as mechanical modes in these ISO static systems.
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,
we want to study new materials, new phenomena, new experiments in these quantum electronic systems.
But there's also many interesting questions in the in the mechanical systems.
You know, so experiments on metamaterials and maybe there are optical electronic plasmonic versions of this kind of phenomena as well.
And of course, an interesting question in both of these settings is the role of interactions and nonlinear narratives.
And those are going to be things which I think are going to keep us busy for for some time to come.
So thank you very much.