Great.
Fantastic. So I'm going to.
I'm here to talk to you a little bit about what's changed in this field in the last
five or so years, and sort of convey why this is still an active area of research.
So just to summarize what our colleagues of my colleagues have told you so far.
So Shivaji started out by telling us how topology has really changed, how
we view phases of matter,
both in how we classify phases and how we understand their exhibitions.
Steve, I've put the disclaimer he should have put on there
that says that these topological ideas could even be useful in certain contexts,
particularly in terms of anyons.
But, you know, in spite of Microsoft's billion dollar investment, I
Oxford Physics does not endorse you're putting any money into any quantum
computing that way.
So just to cover our bases.
So what I
a lot of what Steve said, somebody said, you know, these ideas are recent.
They are recent.
The recent on the timescale of physics you see in textbooks.
But they're not that recent in the timescale of the physics
that people write papers on.
So if you think about it, a lot of these ideas were already there.
And in the, in the ether in the 1990s and 2000.
So a question is, why might this be still something
that's a field of active research that Steve and I, for instance,
spend our days thinking about some fractions of our days
thinking about it when we're not doing admin.
And why in
2025 is this more interesting and much more interesting than 2015?
So what I'm going to try and tell you
is a little bit about the sort of very recent history
and what happened between 2015 and 2025 that made this interesting.
And, you know, to tell that story, I want to tell it to you in sort of three
chapters or three ideas and how those three ideas
kind of connect these, these what Shivaji and Steve talked about.
And the three words I'll use are topology correlations and tune ability.
And in some sense, tune ability is the thing that's really changed.
And I'll try and explain why that changed in recent years.
And so what I want to get across is how does the sort of original,
the vanilla version of the Quantum Hall effect embody these ingredients?
Because that will give us an idea of why taking those ideas and making them work in
other systems was hard. And what changed?
And I want to try and explain why that's challenging
and how people actually managed to do this.
So that's the story I'll try and take you through today.
Okay. So
going back to basics, this is sort of a classic exercise
in that we give our second year quantum mechanics students.
We ask them what happens to the quantum mechanics of an electron
moving in a magnetic field if it was forced to be in two dimensions?
So famously, there's a Lorentz force.
So classically,
instead of moving in straight lines, electrons will form circular orbits.
But quantum mechanics tells us that periodic orbits must be quantized,
and there's an elementary way of quantizing it, which basically thinks
about Arno boom phases that the electrons experienced.
So, you know, there's a calculation you would, you know, work through it.
Your tutor would kind of check your H bars and complain about various factors.
But in the end, if you did it all right, what you should find is that instead
of our happy p squared over two M free particle dispersion,
you get a series of energy levels that look bizarre
because these look like the energy levels of a harmonic oscillator,
but in one dimension.
And what happened is that one dimensions worth of kind of motion of the electrons
went into these levels being very, very degenerate.
And roughly a way to think about what happens in this kind of problems
is that if we think about the particle moving around,
we can actually calculate the the phase it accumulates.
And I don't know, but in Bohm tell us that that phase is some set of fundamental
constants here in CGS line multiplied by the flux through that unit.
And if you work this out, this turns out to be dimensionless.
And when this angle when this, this phase is two pi, it turns out
one can think of that as defining a new unit cell for the particle.
So, you know, in free space there's no unit cell.
All all coordinates are created equal.
But once you put on a magnetic field you have to go around sort of a distance.
You fuzzy out space by that amount.
And any kind of periodicity can only happen on that scale.
And so that roughly is the area occupied by a single quantum of flux.
So h c over E if you put in your units and do dimensional
analysis has the same units of magnetic flux.
So this is a way of counting.
And it also defines a length scale.
And that length scale will come back to you in a little bit.
So what happens when we do this is that you get an interesting problem.
Because if you take a sample
with magnetic field B, an area A, well you can count out,
you can compute the total flux that's just multiplying these two numbers.
And you can parcel it out into flux quanta.
And that defines a number which tells you
how many unit cells have I created in this little sample.
And each of these levels has a degeneracy that corresponds
to counting a whole sample.
And partitioning it into these many unit cells.
And if you think about it,
a area is an extensive quantity in two dimensions,
because if I double the sample, the area will double.
That's what we mean.
And so that means that there's
this huge degeneracy which if you again go back to your,
you know, second year quantum mechanics,
that means that you can have a very painful afternoon
because degenerate perturbations theory is horrible.
So it turns out that there's a useful figure of merit.
This is the number that Steve was using.
And I don't know if he actually mentioned that he was using it,
but you just count up if I give you electrons.
Electrons are fermions. No anyons yet.
And so if I put a
fermion in one of these unit cells, it won't allow another fermion to come in.
So I can count up the number of those flux quanta in my sample
that have an electron or don't have an electron,
and that tells me the fraction of the magnetic
unit cells in the sample that are filled with electrons.
So this means that there's a nice way of dividing our problem when it's an integer.
Well, I just fill in.
So let's imagine I have an imaginary system
with nine flux quanta, and I give you nine electrons.
So this ratio is nine over nine which is one.
This is just a unique way to fill nine electrons in nine boxes,
because you can't put a second electron at any of these boxes.
And so that's nice because I don't have to even worry about anything.
I've got a unique ground state, so I've solved a many body problem.
There's no difficulty here. I don't have to care.
The problem comes in when I go to fractional new.
So for instance, if I had three electrons in those nine boxes,
I've just written down four of the different ways you could arrange them.
But in fact I counted.
And there are 80 more that you can do.
And the way you can see that is nine. Choose three, right?
But of course you have to be even more careful because electrons are fermions.
So you actually have to worry about these things.
Think about the appropriate fermionic statistics, solve some big matrix.
And in fact, you know, a colleague of mine in grad school
said, when you know, all this theoretical physics
is just huffing and puffing and then you diagonalize a big matrix.
And the problem here is that the matrices are very big indeed.
And so, in fact, what you do is you throw up your hands and guess.
And that's really what Laughlin did to win the Nobel Prize
is, in fact, I guess, an answer,
more or less for this problem of three electrons in nine boxes.
So the magic is, of course, that we have physics that comes out of this, that
if you take insulate, you take a system which realizes this physics.
Of course, the N is now in the scale of Avogadro's number.
It's a gigantic scale, and you're putting lots of electrons in.
And what you see is that you see that what you measure is something pretty mundane.
You measure longitudinal and transverse conductivities.
And the big thing to take away from a plot,
the two things take away from this plot is that every time you see a plateau
in the whole conductivity, you've just seen a different phase of matter.
So I can barely can't even count the number of phases of matter on this plot.
And the other remarkable thing is I'm tweaking between phases of matter.
I'm toggling between them by just dialing a knob in my experiment
that changes the electric field on a sample.
So I'm changing a voltage and I'm going between different phases of matter.
So that's a pretty amazing plot if you think about the effort,
once you've made the sample, the effort required to go to different,
different new physics is quite, quite great.
So what I want to do is unpack the ingredients that went into this plot.
So the topology correlations and the tune ability and contrast
what we would get if we wanted to realize this in a different platform
than this very exotic setting of electrons in a high magnetic field.
So that's the goal for what I want to do next.
So the first ingredient is topology.
And you know this in a if you go back to your third year
condensed matter course, then you know, Steve has a very nice textbook.
And if you open that textbook you see that electrons and solids,
satisfy the Schrodinger equation, but with a periodic potential.
And Shivaji pointed this out in this talk.
But I'll remind you that back in the 1930s, Felix Bloch
pointed out that, you know, the solutions to this problem are not plain waves,
but they're almost plain waves.
They look like they have a plane wave piece and a periodic function.
And so the one trick, though, is that
while plane waves can have any momenta, these, these states can only have
certain momenta that live within this object called the Brill one zone.
And the thing to take away is that these energy levels now split up.
So you get these things called energy bands.
So they have a well-defined energy momentum relation.
It's no longer as simple as e of k equals p squared over two m.
If p p squared over two m except near extreme points of the band.
But away from that you see that there are gaps that separate these bands.
And the other thing is that this object, this thing that replaces the momentum,
is no longer a label that's allowed to freely wander to minus or plus infinity.
It has to live on in this one dimensional example, in a circle
and in higher dimensions, on more complex and effectively on a Taurus.
Right.
And so these objects mean that your momentum has now become a weird,
something different from what you expect in free space.
And so this is the physics of energy bands.
This looks very different from the physics of Landau levels.
So how do we connect them?
Well, that was mentioned by about I won't go over the details of it,
but it turns out that
when you think about these energy bands, they can also have topology.
And this is the thing called a Chern insulator, which, you know,
it was understood the first model of this was written down by Duncan Holden.
And the kind of understanding of this
as a topological object was pioneered by David Paulus.
And really they won the Nobel Prize in 2016 for their work.
And so the basic idea is that these, you know, what you want to take away is
these are ways of having your cake and eating it,
to having the sort of topology of a lambda level,
but still having this sort of block like states of electrons
in, where we associate with moving and periodic potentials.
And so the, the thing that characterizes by a single number,
an integer called the Chern number.
And if you fill up one of these bands, you get a response.
So that would be an insulator.
Field bands give you insulators
and that response would be quantized in units of E squared over h.
And in those units the response is just given by the Chern number.
So these realized the lattice version of what we were talking about
when we were talking about.
And just as an aside, where does the topology come from?
That's another story.
So if we think about the kind of picture that's sketched over here,
what Haldane was pointing out is that you could have
a very complicated magnetic field that averaged to zero in a unit cell,
but was non-zero in these regions A, B, and C.
So you could have some very complicated magnetic fluxes.
So an electron moving
there would go through some very weird pattern
in its lambda orbits within that unit cell.
But in the end when all the dust settles, what you find
is that the electronic wavefunction in momentum space wins as it moves.
And that's very similar.
It turns out that winding in momentum space has a very similar effect to,
and a turn off boom phase in a magnetic field.
The thing that I started with,
and the result is that you get this kind of, top topology.
And this kind of physics often arises in systems with strong spin orbit coupling.
You know, that chapter of, atomic physics that you happily glossed over and forgot?
So that was the first ingredient.
So topology, we have a way of taking these top logical ideas of the lambda level
and sticking them into this much more kind of, heavy duty machinery
of band theory, where you have electrons moving in periodic landscapes.
The second ingredient is correlations.
And maybe this is a little bit less a little bit less familiar.
Let me just say what I mean by correlations.
So roughly speaking,
or we're talking about when, you know, when I condensed matter
physicists like Steve Shivaji and myself are thinking about materials, typically
what we're interested in is a competition
between sort of two tendencies, and it's a very coarse level.
You can think of it as the age old story of quantum mechanics.
Is it a wave or is it a particle? Right.
So you can sort of
think about systems like insulators as in some sense being more localized.
Electrons really look like individual charges
sitting in potential worlds and not hopping around.
They don't know that they're moving.
And so we would say that that tends to happen
when things are much more strongly interacting,
either with a lattice potential or with each other.
In contrast, metals and superconductors have a behavior
where they're much more wavelike.
The electron actually explores all the sites in an individual electron,
zipping around all over the sample.
It doesn't stay tied to one site, and any material you give me
will lie somewhere on a continuum of, you know, where it sits.
It's never going to be ideally in one limit or the other.
So that makes the Quantum Hall effect a very special place,
because in lambda levels, the kinetic energy is zero.
So by definition, if I think of this as controlling interaction strength, well,
you know,
one of the first things your tutor will tell you is dimensionless numbers.
The only thing that mean anything.
And so the dimensionless thing we should look at
is the strength of interactions of the kinetic energy.
But the kinetic energy in the quantum Hall effect is zero.
There's no kinetic energy.
All states are created equal.
And so Landau levels intrinsically, if you sit in the middle of a Landau
level, are infinitely strong interactions, which is why the problem
was so hard to solve.
So you're in the strong coupling limit.
Well, that looks very difficult.
It turns out that we have, after 30 years or so good experience with what happens,
that strong coupling, like we solve any problem.
The extreme limits are always easy.
Chern bands and topological bands in general.
Any bands are different because they have a kinetic energy.
You've changed it a little bit, but the electron still moves.
They're not sitting around stationary, and so they're an intermediate coupling,
which is a fancy way of saying I don't know what to do, right?
They're just difficult because you don't have you here.
You can think about,
well, if I turn on interactions, I perturb around free particles.
Here you say, oh, if I turn our interactions
I perturb around electron sitting on sites here, I don't know what the right basis
is or what to do.
So this is why these kinds of problems are difficult.
And typically when you hear terrorists throw up their hands
and say and argue for years, that's really what happens.
And just to say that you can ask the same question,
is it possible to get Chern insulators that look very much like the lambda level?
Can we play games and take that kinetic energy and make it go away?
So, you know, about 15 years ago now, various groups came up with the idea
and it turns out that it was an idea
that was sitting around just waiting for somebody to do it.
It's not that difficult to think about.
You can play with the the way electrons move
between sites in a unit cell so that you make the Chern band.
So this is a topological band.
But notice that while this band up here is doing a lot, if the electrons were here,
it looks really like that kinetic energy is not doing anything.
Whoops.
And also looks like my laptop is not doing anything.
So something about this room really doesn't
like us getting to the right answer.
So yeah, so it turns out under the right conditions and it actually took some time.
Shivaji and I thought a little bit about what
those right conditions should be at, along with a postdoc at Oxford,
Rahul Roy, who actually did a lot of work on these topological ideas.
It turns out under the right conditions you can more or less map the problems of,
of Chern band with very weak, with a very weak dispersion to a Landau level.
So it looks really like you can get pretty close to what the experiments.
What we would like to get something which looks like it has the topology
of a chain band lives in a crystal, so there's no need for a big magnetic field.
The physics came from electrons moving in some kind of complicated
crystal potential, and that has strong interactions.
And there was lots of subsequent theory.
But it turns out while terrorists can write down flattening bands
as just an equation, it's very, very hard to do an experiment.
So that kind of made progress, you know, held up progress for a very long time.
And that brings me to the third ingredient.
What would you like to be able to do in order to do an experiment?
You don't just want to create some particular material.
You want to be able to tune things.
You want to be able to dial a knob and access the physics that you like.
And this is actually where another challenge comes in.
So, you know, ideally we'd like to tune everything.
If you ask a physicist, what if an experimentalist
comes to a terrorist in US, what do we like to tune?
The terrorist will say everything.
But if you really had to pin me down and say,
what is the one thing I would like to tune?
It would be the density of electrons.
And why that is, is if you think about the difference
being the integer and the fractional quantum Hall effect,
how I went from one to the other is by just changing the electron density here
at a filled Landau level, I added a third of a lambda
levels worth of electrons that change the electron density in the system.
Similarly, if I took block bands, I have a band insulator when I fill bands,
and if I add charge,
then they'll go into the next set of bands, and then I could get a metal,
or if they're strong interactions, I could get a kind of insulating state.
And if these bands had topology, then you might get interesting
topological states like Chern or fractional Chern insulators.
So the key point is that density is what allows me to go
from these integer states, where correlations didn't matter
to these fractional states
where correlations mattered, where anyons live and so on.
So I really need to tune the density to get to these interesting states.
So why is that difficult.
So let's say I gave you a traditional material and said change the density.
What you would actually do is go.
And if it was a three dimensional material
that would be grown in a lab by some very complicated
chemistry building next door, they would grow a sample.
They would try and get the density by changing the kind of stuck
to the exact alloying of two different elements in the sample.
They would get some density.
Then they would go and do it again and get the next point on the plot.
So the time scale for that, and this is a very useful unit called GSI.
Does anybody know what GSI stands for?
It's a very what graduate student.
Yeah, exactly.
So it's measured in graduate student years.
So you know, it depends if you have a lot of grant funding.
That's a very short timescale indeed.
But you know times are tough.
So we need to look for other answers.
It's a two dimensional system.
It's always a different and a continuous way to change the density.
So I'm going to take you back to, you know, something
that probably could have even been given on a on your admissions interview.
So I've give you a parallel plate capacitor
which has charge q and minus q on the two plates and a voltage.
I'm going to give you some numbers.
I'm not sure why it's doing that, but let's see.
Responding roughly like my tutorial students
when I give them this problem.
So the typical numbers
you could have here are a distance of about ten nanometers.
That that's about the figure of merit of what we can do in most experiments.
We can bring do atomically thin layers or, to
planes of a two dimensional electron gas
and these hetero structures to about ten nanometers away from each other.
The typical dielectric constant of these systems is about ten.
And, you know,
if I put much more than about a volt of, potential difference between them,
I start doing chemistry rather than physics
because I start changing the atoms and we don't want to go there.
So it's a simple exercise to see that the charge density,
which is just the total charge over the area,
the total charge is just the capacitance times the voltage over
the area is a simple formula for a parallel capacitor that says
that the capacitance is the area times the dielectric constant over the distance.
So I get this nice formula for the charge density.
And if I put in these numbers, I find that I can push
about an electron in a square of size about ten nanometers.
So if I take this with these, with these kind of experimental constraints,
what I can do by changing the voltage by about one volt is change
the density of electrons
by putting about one electron every side of ten nanometers square.
So we want to ask, how does this compare to the length scales
that we have at hand in a system?
So the first case would be looking at the lambda level.
And I said a length scale would make its appearance again.
So the natural length scale in the quantum Hall effect, remember
if we think about the magnetic unit cell that roughly has a scale lby,
which is this quantity I just put in the numbers for you,
it's obviously magnetic field dependent, but it hits this nice scale
of ten nanometers at about ten, Tesla at seven, Tesla
and it goes up as the square root of B.
So if I, you know, double this
this would go up to about you know, it would go down a little bit.
So it would be about seven nanometers
or eight nanometer, seven or around seven nanometers at 15 Tesla.
And in fact, that plot I showed you, if you look if you zoomed
in, you would see it's about a 15 Tesla field.
And if you look at this axis, I'm changing the voltage.
This is units are in actually volts.
So in fact I'm accessing many different quantum states
just by changing the voltage.
And now you understand why I can do that.
Because I was changing the electron density by about an electron,
every magnetic unit cell as I swept through these things.
So that's the nice figure of merit.
So now let's think about the scale.
If I actually handed you one of these Chern insulators in a crystal.
So the typical lattice spacing in the crystal is a few angstrom.
So let's just make a convenient thing about 0.3 nanometers.
And if you translate that into the density I can tune I can fill up.
You can get to about a thousandth of a unit cell.
That's all I can do
by any kind of experiment that involves getting the sample.
It's very, very hard to push a lot, a push more or less charge onto the system.
So what I'm end up with is that even if we were to go to all the effort
of realizing one of these amazing Chern insulators in a sample,
you wouldn't actually be able to do anything with it.
Without going back to this horrible way of looking at things in graduate student
years of trying to build new,
grow a new sample for every filling, that's the only way you could do it,
even though you have this nice two dimensional system.
So just to summarize, this is the state of play that compares the two things.
So we have Landau levels and we have Chern bands.
Each of them has a notion of a filling which basically counts, you know per level
how what's the fraction of it is filled with electrons.
So that's why we have this thing that distinguishes
between integers and fractions.
And so we got one thing right.
We said, well if we fill a Landau level we get an integer quantum Hall insulator.
You feel a Chern band you get an integer Chern insulator.
These are nice because these are not interacting.
We understand a lot about these things.
But now we see that we can readily tune the filling in one case,
but we can't do it anything appreciable for a realistic crystal
with the spacing of about, you know, a third of a nanometer.
And the
other feature here is that we don't have if you just give me
a Landau level, it comes for free with the fact that it has no dispersion.
So it's intrinsically a very strongly interacting system.
Whereas generically in a Chern band, I don't know that
it's so strongly interacting that I would get these interesting states.
Maybe I do all this work, put them in the interactions,
I'm strong enough, and the system just remains a metal.
That's a lot of work to get a metal, right?
So that's the state of play.
And so our challenge is clear.
We want to get energy bands with these ten numbers.
We want correlations to be enhanced relative to the kinetic energy.
And we want to be able to easily change the filling. Right.
That's the three things we'd like to get.
So hopefully it's clear why we need those ingredients.
So let me give you the Hollywood version of that history.
So what happened next.
So you know this is the situation in 2013.
You know theory is going far ahead of experiment.
The reason you can tell this
in experimentalists is they're wearing a tie
even when they're, you know, relaxing at home.
So, you know, they're very distinguished.
2013 to 2018, because they went to experiments,
theorists were doing a lot of very, very esoteric questions, ignoring experiment.
In the meantime, experimentalists were working very, very hard.
And so to come to the natural end of that
story, around 2018, there was new physics.
Right.
And I want to tell you about that new physics and the time
that's left and think I have about 20 minutes left.
So that's roughly right. Great.
Okay.
So the new physics is something called the Moray effect.
And so I went onto the internet and looked for a figure.
And you get this rather nice piece of poetry which says, you know,
which is, you know, making a riff on an old song by Dean Martin.
Right.
So the Moray phenomenon is this sort of remarkable thing.
And, you know, it's sadly often capital ized because it sounds like
it's the name of a person.
The actual story is really interesting.
It's a word that comes from the textile industry.
So Moray Fabrics actually has a long etymology.
All the way from Arabic to English and then via French, back to English.
And what it is basically is that two things.
If I were wearing stripes and you went back and watch this video,
which I doubt you would do,
you would see actually that because
I'm being recorded on a screen with some refresh rate.
Often what you can see is that when people have two screens
or when they have striped patterns and they're being interrogated
at different frequency, you can see this weird pattern of, beating interference.
And so that's something that often comes in
and that's sort of a personal experience many of us have with Moray patterns.
So what it is, it's really a very simple idea.
If you have two periodic structures
and for whatever reason, they have slightly in one dimension,
if they have slightly different, frequency is or in two dimensions,
as you'll see in a minute, they have slightly different orientations.
Then you can actually get a certain type of interference pattern.
So let's see how that works.
So let's just take two waves.
So I'm taking two waves which have a very so this delta is much smaller than k.
What I'm going to think about.
So let's do waves that very slightly differ in their
you know, they're slightly offset with respect to each other.
And you can see that the red and the blue and I add them up.
And what do you see
coming out of this is this potential that looks awfully like it.
Lot like it's periodic.
It isn't really.
It turns out that it's only periodic.
If somehow there's some kind of
rational relationship between delta and K, you can think about this function
and see if it's really periodic.
And you can see that unless the peaks and troughs of what
delta wants to do exactly line up with the peaks and troughs of K,
this won't be really periodic, but gosh, it looks really very close to periodic.
So it's faking me into thinking it's periodic.
And so this is the moiré effect.
When you take two things which take two periodic functions
and add them up, and they have a very slightly offset
frequency, then you get an emergent approximate periodicity, which is much,
much, much bigger than the microscopic periodicity of either of them.
Right.
So they both started out with a periodicity that's about two pi over k.
But this sort of emergent long wavelength
periodicity is something like two pi over delta, which is really big.
So that's an emergent length scale.
So this is all I can do in one dimension.
But if you're in two dimensions you can do something a little bit more exciting.
So let me just pass that out so you can see for yourself how that works.
So you can make two dimensional
atomically thin materials and you can actually play with those.
So pass these up since because that side
and you can slap layers on if you can just pick one
and pass them up.
You can take two atomically thin
layers, place them on top of each other and nobody.
It turns out that the kind of atomically thin materials like graphene,
one reason you can make them is because
they crystallize two very strong bonds in two dimensions.
But the interactions between the layers are held together
are only Vander Walls interactions.
They're very, very, very weak.
So it turns out the way people first made graphene was remarkable.
What they did was they took a piece of Scotch tape,
stuck it on something that was effectively a very, very pure
piece of pencil lead and pulled it off and just kept doing this.
And if you do that enough times, you peel off an individual atomically thin layer.
Right.
But you can do this twice and by a lot of very hard work.
And I don't want to this is exactly what the hard graft of experimentalists.
So my friend Andrea Young was one of the first people who did this.
It was mentioned in Steve Stock.
And he roughly says, it's like you had a kilometer long chopstick in each hand.
You're taking a layer of clingfilm
and trying to rotate it another layer of clingfilm,
all while not being able to see what you're doing right.
So that's roughly what people did.
But they could do that.
They could take two atomically thin layers, align them close to each other,
and then just give them a little bit of a twist.
And so when you do that, you get patterns like this.
Right.
So the left is two kind of stripy patterns.
The right is what happens when you take two hexagonal lattices and rotate them.
And there's a little bit of algebra you say imagine
each thing had a periodicity that was cosine k dot x in each direction.
Well the angle between the two vectors, they have the same length,
but they're rotated relative to each other.
And if you think about the difference between these two vectors
at small angles, it's
just the size of the vector times theta, sort of basic trigonometric calculation.
And so if you think about the fact that the original periodicity
was the lattice scale, which is the previously advertised
0.3 nanometer scale, that number was cleverly chosen
because it is roughly the lattice spacing of, graphene.
And if I pick an angle of about a degree and a half,
it turns out that this Maryland scale is about that magic number of ten nanometers.
So what it means is, if I made one of these samples
and used it as one plate of my capacitor, that has a potential
that's exactly match to what I can put
push charge on and off at the one charge per unit cell level easily.
Right. So I can do that.
So of course I'm not doing anything close to putting one
charge per microscopic cell.
What I'm doing is pushing off
one one electron in every one of these yellow squares.
I can make one of these yellow little shaded regions.
I can pop an electron on and off.
So it's a very, very low density of electrons.
And normally there wouldn't be a potential for it to push against.
But by twisting this lattice I've created a new potential.
It's a much, much lower energy scale than the original potential.
So while the original answer would be a long run electron volt
or a 10th of an electron volt scale,
we're down now to the electron volt scale because a much, much lower energy scale.
But if I do a very sensitive experiment, I can measure things of that scale
so that I'm now going to be playing
with a periodic potential that sits at that scale.
And another fact is that just as the magnetic field modifies
the kinetic energy of free electrons, the fact that you have this potential
turns out to give you flat bands with a very small kinetic energy.
And the great thing about this is,
and I'll come back to this in a little bit, there's a huge number of,
kinds of two dimensional materials.
Graphene was the first discovered, but they're now a huge the libraries of them.
I'll come back to that. So it means that there's essentially
combinatorial possibilities for new physics.
But the hydrogen atom of that new field is basically taking
two graphene layers and twisting them.
And so there's some extra details because graphene famously does not have electrons
that look like free particles, they look more like Dirac particles.
And that actually gives you some extra richness,
which is noted by these two gentlemen, Raffi Spritzer and Alan McDonald.
And they actually came up with a way, it turns out, to actually figure out
what this lattice does.
This new emergent super lattice does is not easy.
You have to figure out new ways to calculate with them.
And they did that.
And what you can I hope what you can see here is by playing around with the angles.
You know that one year, that 1.25 degrees,
what you see is a very, very flat band there.
So technical reasons
why they're actually eight bands in there,
because electrons come with different spins
and they have other kind of labels on them.
But you get these eight energy bands.
And so from what I've told you so far, what you should be thinking about
is that, well, I've got some boring insulator here.
Relatively boring, because after all, I just filled up these bands.
It looks like the insulator in my textbook
because I just have filled bands and then a gap to the next band.
I should get a boring insulator here.
And if you count those boring insulators happen
at fillings of minus four.
And for why there fillings of minus four and four
is that graphene has some natural electron count, so minus
four means I pull away four electrons in every one of those cells.
Plus four means I add four electrons to each of those cells
so I can look at things adding and subtracting charge to that system.
But if I'm inside there, I have a really interesting problem
because essentially the kinetic.
So this the kind of the width of this band, the thickness and energy
gives me a figure of metric for the kinetic energy.
And actually the Coulomb interactions are much, much bigger than this.
So it means that the electrons here are very, very strongly correlated.
Which means if I put charge on there,
anything I see should be coming out
as the result of the electrons talking to each other,
rather than they're talking to the super lattice.
And so the
group of Pablo Hurler who wrote MIT did this in about 2018.
And in fact, what they found was that, yes, you see, the boring
and B ice for band insulator may as well be for boring insulator for this thing.
So there's nothing interesting happening there.
But what they saw was something very surprising.
So you notice that there two.
So what this curve is, is just showing you how conducting the sample is
as I tune the gate voltage, as I tune push charge on and off the system.
And what you see is while it's
really an insulator here and here that I would have expected, that just came
from this calculation of electrons talking to the weird moiré lattice.
But what happens here are two new insulators that nobody expected.
Why is that insulating?
It must be as a result of electron electron interactions.
So it turns out
there are about 100 different proposals for what that insulator was.
And sort of by luck or good judgment,
Steve and I managed to propose the correct one.
And that was confirmed in experiments a year or so ago.
And there's another surprise, which is a surprise
that still remains to be explained.
So if you take those insulators and move a little bit away from them,
you get a superconductor.
Nobody expected that.
Nobody expected graphene to become a superconductor in this situation.
So you've taken an insulator, you've added a little bit of charge,
and somehow, magically, the added charge becomes superconducting.
We don't know why it becomes superconducting.
There's a huge debate in the literature over this.
We don't know what that superconductor is,
whether it's like the old superconductors that people discovered in the 19th.
You know, that people explain successfully in the 1950s
or the new high temperature superconductors.
We don't know what the mechanisms are.
This is a big area of research that lots of people are trying to understand,
but and there's a whole bunch of other phenomena and some hints of topology.
But it turned out that in graphene it was very, very, very hard to get these
topological states to reliably.
And in the end, the only way to get them reliably was to go back to the old idea
of turning on a little bit of a turning on a magnetic field and teasing them out.
So that wasn't very satisfactory, because after all, the way we went into
this was getting rid of the magnetic field entirely and trying to do this
without a big magnetic field, without all these exotic effects.
So we have a nice system,
but only tantalizing hints of topology, and so can we do better.
So indeed, the answer is that we can.
And the reason that we can do better is because of the fact
that there's this combinatorial richness to this new class of platforms,
you know, old material science.
You needed to know a lot of chemistry because you needed to know, you know, what
things will combine with other things to form, you know, good compounds.
How would those compounds crystallize?
Here are the nice idea here is that you have these 2D materials
and you can play combinatorics instead of chemistry.
You can stack the materials on top of each other.
You can twist them. You can pick different materials.
You can try and use different place the resulting thing on a different
substrate to change the nature of interactions between electrons.
And so if you look at the library of 2D
materials, there's actually a lot of them that can be Scotch taped.
Of course, to get the best samples, you do something different than scotch tape.
You do chemical vapor deposition or something like that.
But still, the Scotch tape idea is pretty cool.
And so there's a whole list of them right there.
And so I'm not going to go into any of them except one.
So there's a very important
class of things called transition metal DHL coordinates tm, MDS.
There's molybdenum disulfide, which I think for many years
is used as a lubricant or something like that.
It turns out to be a very important compound for a lot of these experiments.
And there's a whole family and I'm going to focus on these.
And so what these are it doesn't look like this in this rendering.
But they also have a honeycomb lattice.
That's it looks a little bit different than graphene.
But if I looked at it on the 2D plane it would have some honeycomb like pattern.
And it's made up of the unpacking the name.
It's made up of transition
metals like tungsten, like molybdenum, tungsten or hafnium.
And a child.
A Charcot gene is a fancy word for something in the same row
as sulfur in the periodic table.
So sulfur selenium or tellurium.
So the characterized by this rather neat fact that in any individual layer
they just have they have something that looks like a parabolic dispersion.
You don't have to worry too much. And so it looks like we did all of this.
It's like it's not as rich as graphene, an individual layer,
because graphene, after all, had this fancy Dirac electron floating around.
None of that happens here.
Instead you have this just
it looks as though a apart from some label that the mass is a little bit different.
It looks as though you could just
start off with ordinary electrons and free space.
But what's nice about it is that if you look, if you now take one layer
of this TMD and place it another layer, now electrons want to tumble
between the layers and the potential that the two electrons, that electrons
moving in this combined system see are because of that tunneling.
But that tunneling takes a particular form.
So this delta captures the tunneling between the layers.
And that essentially all that tunneling ends up doing is
there's also a different potential in the layer.
So that delta so sigma is some Pauli matrix.
I won't write it down that.
Just think of it as, you know, just like I have spin.
There's ups and downs, but I can represent it as a spin.
Imagine Sigma Plus is in the top layer,
Sigma pointing down is in the bottom layer.
And Sigma in the plane would be you're in one, you're in some superposition
of the two layers with some definite phase.
So it turns out that
this one second.
What you find
is that this delta is like a period.
It's a periodic field.
It's got the periodicity of this emergent moiré supercell.
And that's like a Zeeman field that tells you which layer you want to be in.
So the electron really wants to be in the layer.
It's told to be by that.
And the kinetic energy goes along for the ride
because it doesn't care which layer you're in.
Roughly speaking.
And it turns out that near some special angles, the
this delta forms a lattice, which is known as a skirmish on lattice.
So Shivaji sketch what a skirmish on was.
It's this configuration where, where you kind of wrap the sphere.
So this is a kind of busy plot, but what this is saying is
where this thing is yellow, roughly speaking, and the color is dark.
Your electron is entirely in the bottom layer where it's white.
You are entirely in the top layer.
And where it's in between, you have these blue arrows
that tell you where on the equator you point and if you look at this
little diamond here, that's one unit cell.
And if you count the topological index in that unit cell,
following the rules that Shivaji gave you, there's one skirmish in that unit cell.
You go to the next unit cell, there's another skirmish on.
Now the skirmish on is in an abstract space.
It doesn't really spin. It's this layer on.
But it turns out that this has really interesting consequences for the electrons
to see that what you do is do something that we're
all taught to do is always go to a frame in which the problem looks simple.
So we rotate so that we're always pointing in the our layer
coordinate is always pointing along the way we're told to point by delta.
And if you do that change of basis, what you find is that the problem reduces
to looking exactly like you get for electrons in a magnetic field,
like there's a vector potential that electrons see, but there's also some
effective potential landscape that they see.
And each of these things is periodic with the periodicity of the unit cell.
And it turns out that if this has once coming on per unit cell,
that corresponds almost exactly, it
corresponds exactly to having one flux quantum per unit cell in this new problem.
So what you've just done is realize,
effectively a churn band, which is pretty flat.
It turns out that if you the dust settles, that's what you get. So
let's go back to our wish list and see whether we get the thing.
So we have it turns out there's a particular example
that's very nice where the transition metal is molybdenum.
The gene is tellurium.
So molybdenum Telluride and you know, in typical physics fashion,
when you twist them, you put a little T in front.
So it's tmo t two is the fancy word for this material.
And it turns out if you take the angle of about 3.7 degrees, you get a churn band
that has a good gap to all the other bands.
So you can get this topology, you've got the topology in there.
The bands are flat with the kinetic energy.
That's much less than the interactions.
So you've got correlations and there's tune ability.
So the lattice spacing of the scale is about comparable.
It gives you a length of about ten nanometers which is exactly
what we can dial on and off with a one volt potential.
And you can also it turns out that the interlayer potential
can be used to switch on and off the topology,
which is kind of nice because sometimes it's nice.
One of the best ways to show that you've got
something is to show that you can control. Turn it off.
And if you're an experimentalist,
you really like the ability to switch things on and off.
And in fact, when people made the sample.
So this is actually the second experiment that were done on this.
Now they'll come back to the first experiments in a minute.
So what you should see
here is on the x axis of these plots is that interlayer displacement field.
When that zero the system forms the on lattice.
And the blue curve is the whole resistance.
And the red curve is the longitudinal resistance.
So you see the telltale signature of a quantum Hall like response, a zero
longitudinal resistance and a quantized hold resistance at h over E squared.
So you just created an integer Chern insulator.
But if you tweak the displacement field above a critical value,
you turn off this scummy on lattice and bang, everything goes away.
So you've actually controllable, created a quantum state.
You couldn't you couldn't do this in an a lambda level
because you don't have a knob to switch on and off the topology.
Even more remarkably, if you now, you know, work a little bit harder,
you can actually focus on a different density.
So going from here to here is tuning the other knob,
turning a dial on my gate so I can change the charge
and at this point you're partially filling this band and you find
that there's a whole resistance of, three over two e squared.
Or if you translate that into conductivity, that's a two thirds.
That's like a two thirds quantum Hall state.
There's an active search for using these platforms
to get a whole list of, you know, exotic states.
So non-abelian anyons, like Steve talked about something
more exotic states called fractional topological insulators.
But again, one of the really remarkable things of why progress in this field
is exciting is that, you know, if you go back to the quantum Hall effect.
So Steve talked about the work of Swiss armor and Garside,
who are working in gallium arsenide samples.
So for the fraction of quantum Hall effect,
you know, other groups replicated it in gallium arsenide, but for it
to be replicated in another material took about ten years.
So it's a very long time for it to be reproduced in a second material.
So you might have been forgiven for thinking
maybe there's some very special feature to this one material.
Why should we care? Why is it a universal phenomenon?
Whatever a theorist tells me,
the timescale for the fractional Chern insulator to be reproduced was ten weeks,
because roughly so just a few months twisted, maybe two
people did something in a completely different system,
which is taking, you know,
when you do this exfoliation of graphene, you're trying to get a single layer,
but sometimes you peel off five layers at a time.
And somebody said, why don't we just stick that
five layer sample and play with it and see what it does?
And it turns out you can see a whole bunch of fractions
for the same type of experiment.
It turns out that this physics, you know, it's about a year and a half old.
There's still active debate over what's going on.
Nobody understands this. Do any great detail.
Very happy to say that one of my former students is one of the leading.
I'm not involved in this works.
I can say it is one of the leading people in trying to piece together
what's happening in these materials.
So turns out there could be some proposals.
It realizes some new states of matter that are completely different in some ways
from what we had expected from before.
So just to comment a little bit on the future.
So, you know, now that the
tortoise of experiment has led lab us and they're going forward,
theorists are having to come back and think a little bit more
about how to model these states and also how to look for them.
And so one of the things that Steve talked about
is that a signature of the fractional quantum
Hall effect is this idea of any statistics.
And the way you see it in a Landau level is by doing this edge state interferometry
experiment that Shivaji proposed and Steve has talked about today,
it turns out there's a wrinkle of that in these two dimensional materials.
The wrinkle is that,
you know, when you want to do transport measurements, you have to take gold
or something
like that, or platinum and make electrodes and stick them on the sample.
And chemistry doesn't like that.
It turns out that only a couple of groups, I think at this stage, one group
in the world knows how to make good contacts to these MDS.
So what other what?
But the nice thing about a two dimensional system is
you can look at it and you can look at it slightly
more scientifically and put light in there.
So it turns out the first way people looked at it is by some clever
photo luminescence experiments that were allowed.
There's a signature in this which I won't go into that explains that,
you know, if you see a sharp if you see a kind of kink in this photo
luminescence, you should imagine that
there is a fractional there's an insulating state there.
So this got us thinking could we actually just use light to search for ions?
Can we just shine light on a sample and look for that?
You know, one of the things we're actively working on is can you actually measure
fractional statistics by shining light on samples?
And at least in theory, the answer is yes.
So we're working with experimental groups to realize those ideas.
There's just to illustrate that having a new platform
not only allows us to realize all theoretical ideas,
but forces us to think of new ways
of doing things that we thought we knew how to do well before.
So just in summary, since I'm out of time, you know,
by the simple idea of twisting two dimensional materials,
we're able to
get a confluence of three key new routes to physics, of routes, to new physics.
The idea that you want electronic states with some topology,
where electrons strongly interact with each other,
and where you have the ability to actually see any of those phenomena.
And I think the last time and just say, and,
you know, I just gave a particular example of this richness of the setting
to look for these integer and fractional Chern insulating states.
But there's a whole bunch of other things
you can look for, like the superconductors I talked about and so on.
And so I think just to summarize this idea of probing both familiar states,
but also seeking new ones, using the versatility of these platforms
is really increasingly a center of activity in the field of quantum
condensed matter physics. Today.
So let me close with that and take your questions.
So those are both extremely good questions.
So let me answer the first.
So non-abelian anyons.
It's not the dimension of space. It's an interesting point.
So Stephen I didn't quite mention non-abelian.
So Steve talked about anyons.
So anyons are generalizations of statistics
where, you know, instead of getting a minus one when you take a particle around,
when you exchange two particles, you get a phase.
So it turns out that a class of systems where if you take a system of,
you know, frozen anyons and you take one around the you swap them
instead of getting a phase, you get a matrix.
And that turns out to be intimately linked to the fact that when you have,
essentially when you have a bunch of anyons in your sample,
if they're non-abelian, there are many possible degenerate ground states.
And when you do this transformation, you're doing
some kind of adiabatic rotation between the space of those states.
And in fact,
when I came up with this idea of quantum memory, really the non-abelian case
is the one that's very rich for doing lots of quantum protected operations.
So that's why the non-abelian one has this totemic significance.
If you can achieve that, you can do a lot. So that's the first answer.
The second answer is really very, very insightful because in fact,
if you take graphene and try and strain it,
there are things that look a bit like gravitational fields and so on.
It turns out for graphene, what really happens is that effectively
you put on a magnetic vector potential on the sample.
So it's sort of related but not quite the same thing.
Now for more graphene.
It turns out that in fact, when you take two layers and stick them on
top of each other, it may not surprise you that, you know, while things are
sort of happy being twisted, they're not thrilled with the idea.
Right?
And so that puts lots of strains on the sample.
So there's almost never a sample that, you know, lack some kind of strain.
And so, you know, most people just thought of this as an irrelevant complication.
And they solved these problems which didn't have strain in them.
They got all these disparate answers for why these insulators were forming.
And that graphene thing,
I mean, led by our students and postdocs who insisted that this was important.
Steve and I and our group took that quite seriously.
And it turned out once you took that seriously, that I explained
why there were insulators forming. And that's now
the accepted explanation that it plays a very important role indeed.
So it does play a role.
And there are ways to engineer kind of gravitational like phenomena,
but not quite so straightforwardly as that.
It's much easier to engineer electromagnetic phenomena with strain, if you like.
I don't know if there's any,
thing for neuroscience, because at some level, these are,
you know, the kind of things that have most natural connections
between sort of neuroscience ideas and condensed matter of things
which have lots of randomness in them, lots of interconnections
between particles.
This looks much more like a plain vanilla crystal, just engineered
in a clever way, with some other wrinkles that make it different from the crystals
we see all the time. So I'm not sure if there's a direct connection.
My wife is a neuroscientist, so if there was one,
I think it would create a lot more domestic harmony.
But sadly, I'm not allowed to say I'm working on her stuff just yet.
There is some work on trying to look at twisted nanotubes, but,
you know, one of the things that comes about
when you think about it is that, you know, really this two dimensional, twisting,
kind of requires you to be properly in two dimensions.
So when you talk about nanotubes,
it turns out
you can sort of make an argument, but it has to be at some very, very rigid
specific angles. You can sort of have this extra freedom.
The other thing in nanotubes is you can get a moiré
by just looking at the kind of slight layer offset.
Or if you take two different
nanotube materials like something like that, like a copper and a tube and stuff.
The extent to which that's been explored is not
I mean, of course everything's been explored theoretically.
Right? People like to play these games.
People even talk about how to do it in three dimensions,
because you can imagine that you do some optical lattice where you have,
you know, people can engineer, motion of neutral atoms in electron clouds.
When you cool atoms down, you can make optical potentials.
So they move trapped in atomic worlds, and you can do whatever
that's within to some degree of whatever you like.
You can do whatever you like in those samples,
and you can actually mimic what happens when you twist two of them
or in photonic crystals.
So people have done that.
The problem with all those kinds of platforms,
it's hard to engineer strong interactions in them.
So while you can get that, for instance, photonic systems actually getting strong
interactions in a purely photonic system is very, very hard.
So you can get some more like phenomena, but the full richness
seems to be hard to achieve.
It's possible, but it's not obvious how you would do it.
Yeah, that's a great question.
So I'm not an experimentalist. So you know I'm going to say
what I'm going to tell you is my understanding of it.
So there are various ways you could do it.
I think the standard way is you
have you kind of glue these something you kind of hold the samples on to,
you know, there they're various ways to suspend these samples and stick onto them.
There's ways to pick them up so they exfoliate grabs.
So there's this tear and stack.
So you tear the material, you stack it, you lay it down, set it onto something.
You take the other layer and you're using all of these
with two nano mechanical manipulations.
So you kind of know your angle and you can twist it and drop it on there.
Now the success rate of this, you know, fabulous group
is probably one of the best at doing it.
I think they have something like a 60% success rate, something like that.
But typical groups starting out start at something like 2 or 3%.
These are done just at microfilm with Now the actual preparation is not usually
the experiments are often at millikelvin micro Kelvin temperatures.
I don't know what the temperature scale of the preparation is.
And now there are ways to make
you know, one of the things is that originally
we would actually exfoliate samples, but the best graphene is now
chemical vapor deposited.
And so it's better samples and so on.
So the one thing that only one group in the world
does is one of the materials
I put there is hexagonal boron nitride, which is just like graphene,
except the two carbon atoms in graphene are replaced by boron
and one at one site. And,
nitrogen on the other.
That's a material that turns out to be hard to make in the sense
that essentially there's one group in the world that makes it in Japan,
and they supply the entire world essentially with this thing.
So they're these two gentlemen, Takashi Taniguchi and Kenji Watanabe,
who are on every single experimental paper on this material.
If they don't give you a B and you can't do an experiment, roughly speaking.
And so that's sort of that's the big material challenge is getting good.
N and the graphene stuff is assembly. Yeah.
So I should say one last thing.
There is a remarkable new experiment done in the group of Charlie Lanni
who's at the Weitzman Institute in Israel.
What they've done is take an atomic force microscope, which you usually think of
as a very sharp tip that you use to do atomic scale forces.
And they did something very counterintuitive.
They took that and just chopped its head off.
So instead of having a sharp, pointy pyramid,
they have a flat topped pyramid with a kind of triangular profile.
And they can drape a graphene layer on that pyramid.
And it's about, you know, what is it?
I think it's about 200 nanometers by 200 nanometers in each side.
So you've got this flat surface and the graphene sticks pretty well,
and it stays pretty flat.
So we've got another graphene layer on top.
They can bring that layer down and just twist so they can actually twist
in real time and watch all this physics come, come in by just twisting it.
The experiments are just spectacular.
And you can use this for all kinds of things.
So this is the next frontier.
And dozens of groups around the world are trying to make it.
Now the trial has shown it can be done.
Oh that's fantastic.
So actually for graphene, the magic angle is not completely crazy.
There's a calculation
you can do that rests on the fact that graphene has a linear dispersion.
So what you can do is, you know, maybe I'll just do that on the board.
So you have this picture of.
So this is the blue one zone of graphene.
And the thing you need to know about the one zone of graphene
is it has this thing called a Dirac cone at each of these corners.
Let me focus at one corner. This has a linear dispersion.
So if I looked there e would be like VF times.
Okay.
So now imagine I have two such layers.
So I have one rule one zone like this and I have another one.
I always mess this up. So let's see if I can do it.
That's slightly offset There we go.
So I've twisted them relative to each other.
So I have two such cones.
Right.
You can see that I'm in the same frame.
So one is gone.
Theta over two one side one's gone minus theta over to the other side.
So they started out on top of each other.
They moved so in momentum space, if I drew a picture along the line
that joins them, I would have a picture that looks something like this.
Right.
So this point goes here, this point goes there.
And each of these has E going as VFC
okay.
So the two layers, the only perturbation I'm going to put in
is when the two layers talk to each other. We have color.
Excellent.
So let me actually make this work.
So let's say I have let me just redraw these in color.
So the orange one came here.
The blue one came here.
Each of these are contributed by the two layers.
The only potential of the moiré physics is when the two layer each other.
So if I think about a perturbation theory problem, there's not much to do here.
There's not much to do here.
But at this point I have a degeneracy in the two layers.
Talk to each other.
So the minute I switch on a potential
and if you go back to your second year quantum mechanics,
nature abhors a degeneracy.
So you'll just make a gap over here. Right.
And so that is the potential let's call that deeper.
That's the potential between the layers that.
So this gap is of order deeper.
So we want to ask how far this came out.
Like what what was this energy.
This energy is let's call this distance k.
It turns out that that distance k is like
related to theta,
okay.
It's just that it's a very small scale.
It's related to theta.
It's theta times some number that's known for graphene.
Let me just write it that way okay.
So I have to scale I have VF times k d times theta.
That's how high this energy is.
So this is VF k d theta.
And I have d perp.
This comes from chemistry k d comes from chemistry VF comes from chemistry.
Theta is in my control.
So if I set those two equal to each other
and choose theta that way.
So if I just say.
So all these numbers I know what happens
is that the band gets pushed down by as much as it went up.
So twice to rise and gets whacked down by exactly as much as it went up.
So given that I know something from chemistry
about how the two layers talk to each other,
and these two are numbers from a single layer that I know from chemistry,
I can actually predict an angle at where this happens.
And this is remarkably good.
That's actually an exact calculation of the magic angle.
Now, the actual calculation of why things that further terms don't push
things apart again and so on, requires some details is a fancier thing.
But this is this
gets you to within the level of accuracy I was talking about in the talk.
This gives you the right number. Yeah.
Thanks for asking.
This okay.
I think we just found out. So thank you.