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Great.

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Fantastic. So I'm going to.

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I'm here to talk to you a little bit about
what's changed in this field in the last

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five or so years, and sort of convey why
this is still an active area of research.

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So just to summarize what our colleagues
of my colleagues have told you so far.

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So Shivaji started out by telling us
how topology has really changed, how

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we view phases of matter,

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both in how we classify phases
and how we understand their exhibitions.

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Steve, I've put the disclaimer
he should have put on there

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that says that these topological ideas
could even be useful in certain contexts,

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particularly in terms of anyons.

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But, you know, in spite of Microsoft's
billion dollar investment, I

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Oxford Physics does not endorse
you're putting any money into any quantum

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computing that way.

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So just to cover our bases.

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So what I

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a lot of what Steve said, somebody said,
you know, these ideas are recent.

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They are recent.

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The recent on the timescale of physics
you see in textbooks.

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But they're not that recent
in the timescale of the physics

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that people write papers on.

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So if you think about it,
a lot of these ideas were already there.

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And in the, in the ether
in the 1990s and 2000.

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So a question is,
why might this be still something

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that's a field of active research
that Steve and I, for instance,

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spend our days
thinking about some fractions of our days

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thinking about it
when we're not doing admin.

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And why in

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2025 is this more interesting
and much more interesting than 2015?

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So what I'm going to try and tell you

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is a little bit
about the sort of very recent history

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and what happened between 2015 and 2025
that made this interesting.

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And, you know, to tell that story,
I want to tell it to you in sort of three

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chapters or three ideas
and how those three ideas

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kind of connect these, these
what Shivaji and Steve talked about.

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And the three words I'll use are topology
correlations and tune ability.

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And in some sense, tune ability
is the thing that's really changed.

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And I'll try and explain why
that changed in recent years.

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And so what I want to get across is
how does the sort of original,

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the vanilla version of the Quantum
Hall effect embody these ingredients?

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Because that will give us an idea of why
taking those ideas and making them work in

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other systems was hard. And what changed?

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And I want to try and explain
why that's challenging

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and how people
actually managed to do this.

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So that's the story
I'll try and take you through today.

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Okay. So

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going back to
basics, this is sort of a classic exercise

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in that we give our second year
quantum mechanics students.

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We ask them what happens to the quantum
mechanics of an electron

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moving in a magnetic field
if it was forced to be in two dimensions?

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So famously, there's a Lorentz force.

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So classically,

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instead of moving in straight lines,
electrons will form circular orbits.

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But quantum mechanics tells us
that periodic orbits must be quantized,

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and there's an elementary way
of quantizing it, which basically thinks

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about Arno boom phases
that the electrons experienced.

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So, you know, there's a calculation
you would, you know, work through it.

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Your tutor would kind of check your H bars
and complain about various factors.

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But in the end, if you did it all right,
what you should find is that instead

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of our happy p squared over two M free
particle dispersion,

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you get a series of energy levels
that look bizarre

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because these look like the energy levels
of a harmonic oscillator,

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but in one dimension.

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And what happened is that one dimensions
worth of kind of motion of the electrons

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went into these levels
being very, very degenerate.

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And roughly a way to think about
what happens in this kind of problems

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is that if we think about the particle
moving around,

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we can actually calculate the
the phase it accumulates.

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And I don't know, but in Bohm tell us that
that phase is some set of fundamental

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constants here in CGS line
multiplied by the flux through that unit.

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And if you work this out,
this turns out to be dimensionless.

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And when this angle when this,
this phase is two pi, it turns out

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one can think of that as defining
a new unit cell for the particle.

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So, you know, in free space
there's no unit cell.

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All all coordinates are created equal.

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But once you put on a magnetic field
you have to go around sort of a distance.

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You fuzzy out space by that amount.

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And any kind of periodicity
can only happen on that scale.

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And so that roughly is the area
occupied by a single quantum of flux.

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So h c over E if you put in your units
and do dimensional

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analysis
has the same units of magnetic flux.

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So this is a way of counting.

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And it also defines a length scale.

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And that length scale
will come back to you in a little bit.

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So what happens when we do this
is that you get an interesting problem.

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Because if you take a sample

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with magnetic field
B, an area A, well you can count out,

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you can compute the total flux
that's just multiplying these two numbers.

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And you can parcel it
out into flux quanta.

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And that defines a number which tells you

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how many unit cells have
I created in this little sample.

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And each of these levels has a degeneracy
that corresponds

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to counting a whole sample.

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And partitioning it
into these many unit cells.

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And if you think about it,

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a area is an extensive quantity
in two dimensions,

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because if I double the sample,
the area will double.

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That's what we mean.

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And so that means that there's

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this huge degeneracy
which if you again go back to your,

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you know, second year quantum mechanics,

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that means that you can have
a very painful afternoon

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because degenerate perturbations
theory is horrible.

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So it turns out that there's
a useful figure of merit.

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This is the number that Steve was using.

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And I don't know if he actually mentioned
that he was using it,

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but you just count up if I give you
electrons.

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Electrons are fermions. No anyons yet.

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And so if I put a

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fermion in one of these unit cells,
it won't allow another fermion to come in.

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So I can count up the number of those flux
quanta in my sample

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that have an electron
or don't have an electron,

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and that tells me
the fraction of the magnetic

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unit cells in the sample
that are filled with electrons.

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So this means that there's a nice way of
dividing our problem when it's an integer.

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Well, I just fill in.

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So let's imagine
I have an imaginary system

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with nine flux quanta,
and I give you nine electrons.

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So this ratio is nine over nine
which is one.

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This is just a unique way
to fill nine electrons in nine boxes,

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because you can't put a second electron
at any of these boxes.

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And so that's nice because
I don't have to even worry about anything.

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I've got a unique ground state,
so I've solved a many body problem.

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There's no difficulty here.
I don't have to care.

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The problem comes in
when I go to fractional new.

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So for instance, if I had three electrons
in those nine boxes,

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I've just written down four of the
different ways you could arrange them.

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But in fact I counted.

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And there are 80 more that you can do.

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And the way you can see
that is nine. Choose three, right?

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But of course you have to be even more
careful because electrons are fermions.

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So you actually have to worry
about these things.

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Think about the appropriate fermionic
statistics, solve some big matrix.

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And in fact, you know,
a colleague of mine in grad school

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said, when you know, all this
theoretical physics

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is just huffing and puffing
and then you diagonalize a big matrix.

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And the problem here
is that the matrices are very big indeed.

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And so, in fact, what you do
is you throw up your hands and guess.

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And that's really what
Laughlin did to win the Nobel Prize

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is, in fact, I guess, an answer,

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more or less for this problem
of three electrons in nine boxes.

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So the magic is, of course, that we have
physics that comes out of this, that

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if you take insulate, you take a system
which realizes this physics.

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Of course, the N is now
in the scale of Avogadro's number.

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It's a gigantic scale,
and you're putting lots of electrons in.

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And what you see is that you see that what
you measure is something pretty mundane.

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You measure longitudinal
and transverse conductivities.

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And the big thing
to take away from a plot,

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the two things take away from this plot
is that every time you see a plateau

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in the whole conductivity, you've
just seen a different phase of matter.

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So I can barely can't even count the
number of phases of matter on this plot.

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And the other remarkable thing
is I'm tweaking between phases of matter.

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I'm toggling between them
by just dialing a knob in my experiment

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that changes
the electric field on a sample.

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So I'm changing a voltage and I'm
going between different phases of matter.

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So that's a pretty amazing plot
if you think about the effort,

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once you've made the sample,
the effort required to go to different,

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different new physics is quite,
quite great.

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So what I want to do is unpack
the ingredients that went into this plot.

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So the topology correlations
and the tune ability and contrast

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what we would get if we wanted
to realize this in a different platform

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than this very exotic setting of electrons
in a high magnetic field.

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So that's the goal for what
I want to do next.

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So the first ingredient is topology.

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And you know this in a
if you go back to your third year

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condensed matter course, then you know,
Steve has a very nice textbook.

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And if you open that textbook
you see that electrons and solids,

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satisfy the Schrodinger equation,
but with a periodic potential.

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And Shivaji pointed this out in this talk.

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But I'll remind you
that back in the 1930s, Felix Bloch

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pointed out that, you know, the solutions
to this problem are not plain waves,

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but they're almost plain waves.

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They look like they have a plane
wave piece and a periodic function.

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And so the one trick, though, is that

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while plane waves can have any momenta,
these, these states can only have

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certain momenta that live within this
object called the Brill one zone.

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And the thing to take away
is that these energy levels now split up.

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So you get these
things called energy bands.

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So they have a well-defined energy
momentum relation.

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It's no longer as simple as e of k equals
p squared over two m.

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If p p squared over two m
except near extreme points of the band.

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But away from that you see that
there are gaps that separate these bands.

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And the other thing is that this object,
this thing that replaces the momentum,

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is no longer a label that's allowed to
freely wander to minus or plus infinity.

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It has to live on in this one
dimensional example, in a circle

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and in higher dimensions, on more complex
and effectively on a Taurus.

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Right.

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And so these objects mean
that your momentum has now become a weird,

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something different
from what you expect in free space.

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And so this is the physics of energy
bands.

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This looks very different
from the physics of Landau levels.

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So how do we connect them?

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Well, that was mentioned by about
I won't go over the details of it,

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but it turns out that

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when you think about these energy bands,
they can also have topology.

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And this is the thing
called a Chern insulator, which, you know,

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it was understood the first model of this
was written down by Duncan Holden.

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And the kind of understanding of this

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as a topological object
was pioneered by David Paulus.

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And really they won the Nobel Prize
in 2016 for their work.

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And so the basic idea is that these,
you know, what you want to take away is

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these are ways of having your cake
and eating it,

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to having the sort of topology
of a lambda level,

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but still having this sort of block
like states of electrons

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in, where we associate
with moving and periodic potentials.

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And so the, the thing that characterizes
by a single number,

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an integer called the Chern number.

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And if you fill up one of these bands,
you get a response.

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00:10:02,560 --> 00:10:03,880
So that would be an insulator.

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00:10:03,880 --> 00:10:05,840
Field bands give you insulators

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00:10:05,840 --> 00:10:09,320
and that response would be quantized
in units of E squared over h.

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00:10:09,600 --> 00:10:12,960
And in those units the response is
just given by the Chern number.

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00:10:12,960 --> 00:10:16,520
So these realized the lattice version
of what we were talking about

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00:10:16,600 --> 00:10:17,800
when we were talking about.

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00:10:17,800 --> 00:10:20,720
And just as an aside,
where does the topology come from?

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00:10:20,720 --> 00:10:21,840
That's another story.

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So if we think about the kind of picture
that's sketched over here,

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what Haldane was pointing out
is that you could have

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00:10:27,640 --> 00:10:31,320
a very complicated magnetic field
that averaged to zero in a unit cell,

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but was non-zero in these regions A,
B, and C.

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So you could have some very complicated
magnetic fluxes.

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So an electron moving

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00:10:36,960 --> 00:10:39,200
there would go through
some very weird pattern

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00:10:39,200 --> 00:10:41,400
in its lambda orbits
within that unit cell.

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But in the end
when all the dust settles, what you find

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is that the electronic wavefunction
in momentum space wins as it moves.

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And that's very similar.

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00:10:49,280 --> 00:10:52,720
It turns out that winding in momentum
space has a very similar effect to,

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00:10:52,720 --> 00:10:54,920
and a turn off
boom phase in a magnetic field.

230
00:10:54,920 --> 00:10:56,520
The thing that I started with,

231
00:10:56,520 --> 00:10:59,880
and the result is that you get this kind
of, top topology.

232
00:11:00,080 --> 00:11:04,320
And this kind of physics often arises in
systems with strong spin orbit coupling.

233
00:11:04,320 --> 00:11:08,520
You know, that chapter of, atomic physics
that you happily glossed over and forgot?

234
00:11:10,280 --> 00:11:11,600
So that was the first ingredient.

235
00:11:11,600 --> 00:11:15,280
So topology, we have a way of taking these
top logical ideas of the lambda level

236
00:11:15,400 --> 00:11:19,560
and sticking them into this much more kind
of, heavy duty machinery

237
00:11:19,560 --> 00:11:22,560
of band theory, where you have electrons
moving in periodic landscapes.

238
00:11:23,080 --> 00:11:24,840
The second ingredient is correlations.

239
00:11:24,840 --> 00:11:27,840
And maybe this is a little bit
less a little bit less familiar.

240
00:11:28,080 --> 00:11:30,120
Let me just say what I mean
by correlations.

241
00:11:30,120 --> 00:11:31,480
So roughly speaking,

242
00:11:31,480 --> 00:11:33,680
or we're talking about when,
you know, when I condensed matter

243
00:11:33,680 --> 00:11:37,640
physicists like Steve Shivaji and myself
are thinking about materials, typically

244
00:11:37,640 --> 00:11:38,960
what we're interested in is a competition

245
00:11:38,960 --> 00:11:41,920
between sort of two tendencies,
and it's a very coarse level.

246
00:11:41,920 --> 00:11:44,360
You can think of it
as the age old story of quantum mechanics.

247
00:11:44,360 --> 00:11:46,520
Is it a wave or is it a particle? Right.

248
00:11:46,520 --> 00:11:47,040
So you can sort of

249
00:11:47,040 --> 00:11:51,000
think about systems like insulators
as in some sense being more localized.

250
00:11:51,000 --> 00:11:53,440
Electrons
really look like individual charges

251
00:11:53,440 --> 00:11:55,440
sitting in potential worlds
and not hopping around.

252
00:11:55,440 --> 00:11:56,920
They don't know that they're moving.

253
00:11:56,920 --> 00:11:59,080
And so we would say that
that tends to happen

254
00:11:59,080 --> 00:12:01,240
when things are much
more strongly interacting,

255
00:12:01,240 --> 00:12:03,960
either with a lattice potential
or with each other.

256
00:12:03,960 --> 00:12:07,200
In contrast,
metals and superconductors have a behavior

257
00:12:07,200 --> 00:12:08,480
where they're much more wavelike.

258
00:12:08,480 --> 00:12:11,520
The electron actually explores
all the sites in an individual electron,

259
00:12:11,560 --> 00:12:12,880
zipping around all over the sample.

260
00:12:12,880 --> 00:12:16,600
It doesn't stay tied to one site,
and any material you give me

261
00:12:16,680 --> 00:12:19,560
will lie somewhere on a continuum of,
you know, where it sits.

262
00:12:19,560 --> 00:12:22,240
It's never going to be ideally
in one limit or the other.

263
00:12:22,240 --> 00:12:25,240
So that makes the Quantum Hall effect
a very special place,

264
00:12:25,240 --> 00:12:28,080
because in lambda levels,
the kinetic energy is zero.

265
00:12:28,080 --> 00:12:31,760
So by definition, if I think of this
as controlling interaction strength, well,

266
00:12:31,760 --> 00:12:32,600
you know,

267
00:12:32,600 --> 00:12:35,640
one of the first things your tutor
will tell you is dimensionless numbers.

268
00:12:35,640 --> 00:12:37,080
The only thing that mean anything.

269
00:12:37,080 --> 00:12:38,760
And so the dimensionless thing
we should look at

270
00:12:38,760 --> 00:12:41,400
is the strength of interactions
of the kinetic energy.

271
00:12:41,400 --> 00:12:43,560
But the kinetic energy in the quantum
Hall effect is zero.

272
00:12:43,560 --> 00:12:44,760
There's no kinetic energy.

273
00:12:44,760 --> 00:12:46,600
All states are created equal.

274
00:12:46,600 --> 00:12:50,600
And so Landau levels intrinsically,
if you sit in the middle of a Landau

275
00:12:50,600 --> 00:12:53,840
level, are infinitely strong interactions,
which is why the problem

276
00:12:53,840 --> 00:12:55,560
was so hard to solve.

277
00:12:55,560 --> 00:12:57,360
So you're in the strong coupling limit.

278
00:12:57,360 --> 00:12:58,720
Well, that looks very difficult.

279
00:12:58,720 --> 00:13:02,520
It turns out that we have, after 30 years
or so good experience with what happens,

280
00:13:02,520 --> 00:13:04,680
that strong coupling,
like we solve any problem.

281
00:13:04,680 --> 00:13:07,160
The extreme limits are always easy.

282
00:13:07,160 --> 00:13:09,880
Chern bands
and topological bands in general.

283
00:13:09,880 --> 00:13:12,240
Any bands are different
because they have a kinetic energy.

284
00:13:12,240 --> 00:13:14,640
You've changed it a little bit,
but the electron still moves.

285
00:13:14,640 --> 00:13:18,280
They're not sitting around stationary,
and so they're an intermediate coupling,

286
00:13:18,280 --> 00:13:20,880
which is a fancy way of saying
I don't know what to do, right?

287
00:13:20,880 --> 00:13:23,200
They're just difficult because you don't
have you here.

288
00:13:23,200 --> 00:13:24,000
You can think about,

289
00:13:24,000 --> 00:13:27,000
well, if I turn on interactions,
I perturb around free particles.

290
00:13:27,120 --> 00:13:28,720
Here you say, oh,
if I turn our interactions

291
00:13:28,720 --> 00:13:32,480
I perturb around electron sitting on sites
here, I don't know what the right basis

292
00:13:32,480 --> 00:13:33,280
is or what to do.

293
00:13:33,280 --> 00:13:35,720
So this is why these kinds of problems
are difficult.

294
00:13:35,720 --> 00:13:38,040
And typically when you hear terrorists
throw up their hands

295
00:13:38,040 --> 00:13:40,440
and say and argue for years,
that's really what happens.

296
00:13:41,640 --> 00:13:44,600
And just to say that you can ask
the same question,

297
00:13:44,600 --> 00:13:48,760
is it possible to get Chern insulators
that look very much like the lambda level?

298
00:13:48,760 --> 00:13:52,560
Can we play games and take that
kinetic energy and make it go away?

299
00:13:52,560 --> 00:13:56,600
So, you know, about 15 years ago
now, various groups came up with the idea

300
00:13:56,600 --> 00:13:57,720
and it turns out that it was an idea

301
00:13:57,720 --> 00:13:59,680
that was sitting around
just waiting for somebody to do it.

302
00:13:59,680 --> 00:14:02,080
It's not that difficult to think about.

303
00:14:02,080 --> 00:14:05,200
You can play with the
the way electrons move

304
00:14:05,200 --> 00:14:09,240
between sites in a unit cell
so that you make the Chern band.

305
00:14:09,240 --> 00:14:10,800
So this is a topological band.

306
00:14:10,800 --> 00:14:14,240
But notice that while this band up here is
doing a lot, if the electrons were here,

307
00:14:14,400 --> 00:14:17,400
it looks really like that
kinetic energy is not doing anything.

308
00:14:18,000 --> 00:14:18,360
Whoops.

309
00:14:19,880 --> 00:14:21,880
And also looks
like my laptop is not doing anything.

310
00:14:21,880 --> 00:14:24,960
So something about this room
really doesn't

311
00:14:24,960 --> 00:14:26,880
like us getting to the right answer.

312
00:14:26,880 --> 00:14:31,800
So yeah, so it turns out under the right
conditions and it actually took some time.

313
00:14:31,880 --> 00:14:33,360
Shivaji
and I thought a little bit about what

314
00:14:33,360 --> 00:14:36,360
those right conditions should be at,
along with a postdoc at Oxford,

315
00:14:36,360 --> 00:14:39,360
Rahul Roy, who actually did a lot of work
on these topological ideas.

316
00:14:39,520 --> 00:14:44,120
It turns out under the right conditions
you can more or less map the problems of,

317
00:14:44,400 --> 00:14:48,400
of Chern band with very weak, with
a very weak dispersion to a Landau level.

318
00:14:48,400 --> 00:14:52,280
So it looks really like you can get
pretty close to what the experiments.

319
00:14:53,080 --> 00:14:56,760
What we would like to get something
which looks like it has the topology

320
00:14:56,760 --> 00:15:01,120
of a chain band lives in a crystal, so
there's no need for a big magnetic field.

321
00:15:01,120 --> 00:15:04,320
The physics came from electrons
moving in some kind of complicated

322
00:15:04,320 --> 00:15:06,960
crystal potential,
and that has strong interactions.

323
00:15:06,960 --> 00:15:08,720
And there was lots of subsequent theory.

324
00:15:08,720 --> 00:15:11,520
But it turns out while terrorists
can write down flattening bands

325
00:15:11,520 --> 00:15:14,520
as just an equation, it's
very, very hard to do an experiment.

326
00:15:14,920 --> 00:15:19,000
So that kind of made progress, you know,
held up progress for a very long time.

327
00:15:19,720 --> 00:15:21,600
And that brings me
to the third ingredient.

328
00:15:21,600 --> 00:15:24,480
What would you like to be able to do
in order to do an experiment?

329
00:15:24,480 --> 00:15:27,000
You don't just want to create
some particular material.

330
00:15:27,000 --> 00:15:28,320
You want to be able to tune things.

331
00:15:28,320 --> 00:15:31,320
You want to be able to dial a knob
and access the physics that you like.

332
00:15:31,400 --> 00:15:34,400
And this is actually where
another challenge comes in.

333
00:15:34,480 --> 00:15:36,240
So, you know,
ideally we'd like to tune everything.

334
00:15:36,240 --> 00:15:38,080
If you ask a physicist,
what if an experimentalist

335
00:15:38,080 --> 00:15:40,320
comes to a terrorist in US,
what do we like to tune?

336
00:15:40,320 --> 00:15:41,880
The terrorist will say everything.

337
00:15:41,880 --> 00:15:43,720
But if you really had to pin me down
and say,

338
00:15:43,720 --> 00:15:45,920
what is the one thing
I would like to tune?

339
00:15:45,920 --> 00:15:48,280
It would be the density of electrons.

340
00:15:48,280 --> 00:15:50,920
And why that is, is
if you think about the difference

341
00:15:50,920 --> 00:15:53,160
being the integer
and the fractional quantum Hall effect,

342
00:15:53,160 --> 00:15:57,480
how I went from one to the other is by
just changing the electron density here

343
00:15:57,600 --> 00:16:00,840
at a filled Landau level,
I added a third of a lambda

344
00:16:00,840 --> 00:16:04,080
levels worth of electrons that change
the electron density in the system.

345
00:16:04,840 --> 00:16:09,160
Similarly, if I took block bands,
I have a band insulator when I fill bands,

346
00:16:09,360 --> 00:16:10,760
and if I add charge,

347
00:16:10,760 --> 00:16:13,800
then they'll go into the next set
of bands, and then I could get a metal,

348
00:16:13,800 --> 00:16:16,800
or if they're strong interactions,
I could get a kind of insulating state.

349
00:16:17,160 --> 00:16:20,640
And if these bands had topology,
then you might get interesting

350
00:16:20,640 --> 00:16:23,640
topological states
like Chern or fractional Chern insulators.

351
00:16:23,760 --> 00:16:27,480
So the key point is
that density is what allows me to go

352
00:16:27,480 --> 00:16:30,160
from these integer states,
where correlations didn't matter

353
00:16:30,160 --> 00:16:31,320
to these fractional states

354
00:16:31,320 --> 00:16:34,520
where correlations mattered,
where anyons live and so on.

355
00:16:34,720 --> 00:16:38,280
So I really need to tune the density
to get to these interesting states.

356
00:16:38,280 --> 00:16:39,800
So why is that difficult.

357
00:16:40,800 --> 00:16:43,800
So let's say I gave you a traditional
material and said change the density.

358
00:16:44,120 --> 00:16:46,120
What you would actually do is go.

359
00:16:46,120 --> 00:16:47,760
And if it was a three dimensional material

360
00:16:47,760 --> 00:16:50,280
that would be grown in a lab
by some very complicated

361
00:16:50,280 --> 00:16:52,680
chemistry building next door,
they would grow a sample.

362
00:16:52,680 --> 00:16:55,440
They would try and get the density
by changing the kind of stuck

363
00:16:55,440 --> 00:16:59,520
to the exact alloying of two
different elements in the sample.

364
00:16:59,720 --> 00:17:01,200
They would get some density.

365
00:17:01,200 --> 00:17:04,400
Then they would go and do it again
and get the next point on the plot.

366
00:17:04,640 --> 00:17:07,920
So the time scale for that,
and this is a very useful unit called GSI.

367
00:17:07,920 --> 00:17:10,560
Does anybody know what GSI stands for?

368
00:17:10,560 --> 00:17:12,600
It's a very what graduate student.

369
00:17:12,600 --> 00:17:14,400
Yeah, exactly.

370
00:17:14,400 --> 00:17:16,400
So it's measured
in graduate student years.

371
00:17:16,400 --> 00:17:18,800
So you know, it depends
if you have a lot of grant funding.

372
00:17:18,800 --> 00:17:20,640
That's a very short timescale indeed.

373
00:17:20,640 --> 00:17:22,120
But you know times are tough.

374
00:17:22,120 --> 00:17:23,520
So we need to look for other answers.

375
00:17:24,520 --> 00:17:25,800
It's a two dimensional system.

376
00:17:25,800 --> 00:17:30,400
It's always a different and
a continuous way to change the density.

377
00:17:30,560 --> 00:17:32,480
So I'm going to take you back
to, you know, something

378
00:17:32,480 --> 00:17:36,320
that probably could have even been given
on a on your admissions interview.

379
00:17:36,640 --> 00:17:38,640
So I've give you a parallel plate
capacitor

380
00:17:38,640 --> 00:17:41,640
which has charge q and minus q
on the two plates and a voltage.

381
00:17:41,800 --> 00:17:44,160
I'm going to give you some numbers.

382
00:17:44,160 --> 00:17:47,160
I'm not sure why it's doing that,
but let's see.

383
00:17:48,720 --> 00:17:50,600
Responding roughly
like my tutorial students

384
00:17:50,600 --> 00:17:53,160
when I give them this problem.

385
00:17:53,160 --> 00:17:55,120
So the typical numbers

386
00:17:55,120 --> 00:17:58,360
you could have here are a distance
of about ten nanometers.

387
00:17:58,360 --> 00:18:01,440
That that's about the figure of merit
of what we can do in most experiments.

388
00:18:01,440 --> 00:18:05,480
We can bring do
atomically thin layers or, to

389
00:18:06,640 --> 00:18:08,400
planes of a two dimensional electron gas

390
00:18:08,400 --> 00:18:11,400
and these hetero structures to
about ten nanometers away from each other.

391
00:18:11,640 --> 00:18:14,640
The typical dielectric
constant of these systems is about ten.

392
00:18:14,680 --> 00:18:15,320
And, you know,

393
00:18:15,320 --> 00:18:18,560
if I put much more than about a volt of,
potential difference between them,

394
00:18:18,560 --> 00:18:20,520
I start doing chemistry
rather than physics

395
00:18:20,520 --> 00:18:23,080
because I start changing the atoms
and we don't want to go there.

396
00:18:23,080 --> 00:18:27,160
So it's a simple exercise
to see that the charge density,

397
00:18:27,160 --> 00:18:29,520
which is just the total charge
over the area,

398
00:18:29,520 --> 00:18:32,640
the total charge is just
the capacitance times the voltage over

399
00:18:32,640 --> 00:18:36,240
the area is a simple formula
for a parallel capacitor that says

400
00:18:36,240 --> 00:18:39,840
that the capacitance is the area times
the dielectric constant over the distance.

401
00:18:40,040 --> 00:18:42,440
So I get this nice formula
for the charge density.

402
00:18:42,440 --> 00:18:45,160
And if I put in these numbers,
I find that I can push

403
00:18:45,160 --> 00:18:48,720
about an electron
in a square of size about ten nanometers.

404
00:18:48,880 --> 00:18:52,880
So if I take this with these, with these
kind of experimental constraints,

405
00:18:53,040 --> 00:18:56,640
what I can do by changing the voltage
by about one volt is change

406
00:18:56,640 --> 00:18:57,960
the density of electrons

407
00:18:57,960 --> 00:19:01,920
by putting about one electron
every side of ten nanometers square.

408
00:19:02,360 --> 00:19:05,040
So we want to ask,
how does this compare to the length scales

409
00:19:05,040 --> 00:19:06,760
that we have at hand in a system?

410
00:19:06,760 --> 00:19:09,280
So the first case would be looking
at the lambda level.

411
00:19:09,280 --> 00:19:12,000
And I said a length
scale would make its appearance again.

412
00:19:12,000 --> 00:19:15,240
So the natural length scale
in the quantum Hall effect, remember

413
00:19:15,720 --> 00:19:20,040
if we think about the magnetic unit cell
that roughly has a scale lby,

414
00:19:20,040 --> 00:19:23,040
which is this quantity
I just put in the numbers for you,

415
00:19:23,320 --> 00:19:26,480
it's obviously magnetic field dependent,
but it hits this nice scale

416
00:19:26,640 --> 00:19:29,640
of ten nanometers
at about ten, Tesla at seven, Tesla

417
00:19:30,000 --> 00:19:31,560
and it goes up as the square root of B.

418
00:19:31,560 --> 00:19:33,040
So if I, you know, double this

419
00:19:33,040 --> 00:19:35,520
this would go up to about you know,
it would go down a little bit.

420
00:19:35,520 --> 00:19:37,320
So it would be about seven nanometers

421
00:19:37,320 --> 00:19:40,320
or eight nanometer, seven or around
seven nanometers at 15 Tesla.

422
00:19:40,840 --> 00:19:43,720
And in fact, that plot
I showed you, if you look if you zoomed

423
00:19:43,720 --> 00:19:45,960
in, you would see
it's about a 15 Tesla field.

424
00:19:45,960 --> 00:19:48,040
And if you look at this axis,
I'm changing the voltage.

425
00:19:48,040 --> 00:19:50,360
This is units are in actually volts.

426
00:19:50,360 --> 00:19:53,360
So in fact I'm accessing
many different quantum states

427
00:19:53,360 --> 00:19:54,720
just by changing the voltage.

428
00:19:54,720 --> 00:19:56,400
And now you understand why I can do that.

429
00:19:56,400 --> 00:20:00,240
Because I was changing
the electron density by about an electron,

430
00:20:00,240 --> 00:20:02,800
every magnetic unit cell
as I swept through these things.

431
00:20:02,800 --> 00:20:04,680
So that's the nice figure of merit.

432
00:20:04,680 --> 00:20:06,160
So now let's think about the scale.

433
00:20:06,160 --> 00:20:09,160
If I actually handed you one of these
Chern insulators in a crystal.

434
00:20:09,480 --> 00:20:12,280
So the typical lattice
spacing in the crystal is a few angstrom.

435
00:20:12,280 --> 00:20:15,360
So let's just make a convenient thing
about 0.3 nanometers.

436
00:20:15,640 --> 00:20:19,800
And if you translate that into the density
I can tune I can fill up.

437
00:20:20,280 --> 00:20:22,560
You can get to
about a thousandth of a unit cell.

438
00:20:22,560 --> 00:20:23,600
That's all I can do

439
00:20:23,600 --> 00:20:26,520
by any kind of experiment
that involves getting the sample.

440
00:20:26,520 --> 00:20:30,920
It's very, very hard to push a lot, a push
more or less charge onto the system.

441
00:20:31,360 --> 00:20:34,800
So what I'm end up with is that
even if we were to go to all the effort

442
00:20:34,800 --> 00:20:38,000
of realizing one of these
amazing Chern insulators in a sample,

443
00:20:38,040 --> 00:20:39,800
you wouldn't actually be able
to do anything with it.

444
00:20:39,800 --> 00:20:43,440
Without going back to this horrible way
of looking at things in graduate student

445
00:20:43,440 --> 00:20:45,040
years of trying to build new,

446
00:20:45,040 --> 00:20:47,880
grow a new sample for every filling,
that's the only way you could do it,

447
00:20:47,880 --> 00:20:50,480
even though you have this nice
two dimensional system.

448
00:20:50,480 --> 00:20:54,360
So just to summarize, this is the state
of play that compares the two things.

449
00:20:54,680 --> 00:20:56,920
So we have Landau levels
and we have Chern bands.

450
00:20:56,920 --> 00:21:01,360
Each of them has a notion of a filling
which basically counts, you know per level

451
00:21:01,440 --> 00:21:03,920
how what's the fraction of it
is filled with electrons.

452
00:21:03,920 --> 00:21:06,120
So that's why we have this thing
that distinguishes

453
00:21:06,120 --> 00:21:08,160
between integers and fractions.

454
00:21:08,160 --> 00:21:09,760
And so we got one thing right.

455
00:21:09,760 --> 00:21:13,160
We said, well if we fill a Landau level
we get an integer quantum Hall insulator.

456
00:21:13,200 --> 00:21:16,160
You feel a Chern band
you get an integer Chern insulator.

457
00:21:16,160 --> 00:21:18,000
These are nice
because these are not interacting.

458
00:21:18,000 --> 00:21:19,880
We understand a lot about these things.

459
00:21:19,880 --> 00:21:23,600
But now we see that we can readily tune
the filling in one case,

460
00:21:23,600 --> 00:21:26,840
but we can't do it anything appreciable
for a realistic crystal

461
00:21:26,840 --> 00:21:29,440
with the spacing of about,
you know, a third of a nanometer.

462
00:21:31,200 --> 00:21:32,160
And the

463
00:21:32,160 --> 00:21:35,240
other feature here is that we don't have
if you just give me

464
00:21:35,240 --> 00:21:39,080
a Landau level, it comes for free
with the fact that it has no dispersion.

465
00:21:39,080 --> 00:21:41,920
So it's intrinsically
a very strongly interacting system.

466
00:21:41,920 --> 00:21:44,760
Whereas generically in a Chern band,
I don't know that

467
00:21:44,760 --> 00:21:47,880
it's so strongly interacting
that I would get these interesting states.

468
00:21:47,880 --> 00:21:50,560
Maybe I do all this work,
put them in the interactions,

469
00:21:50,560 --> 00:21:53,520
I'm strong enough,
and the system just remains a metal.

470
00:21:53,520 --> 00:21:55,600
That's a lot of work
to get a metal, right?

471
00:21:55,600 --> 00:21:58,280
So that's the state of play.

472
00:21:58,280 --> 00:21:59,960
And so our challenge is clear.

473
00:21:59,960 --> 00:22:02,520
We want to get energy bands
with these ten numbers.

474
00:22:02,520 --> 00:22:05,800
We want correlations to be enhanced
relative to the kinetic energy.

475
00:22:06,000 --> 00:22:08,760
And we want to be able to easily
change the filling. Right.

476
00:22:08,760 --> 00:22:10,280
That's the three things we'd like to get.

477
00:22:10,280 --> 00:22:12,600
So hopefully it's clear
why we need those ingredients.

478
00:22:12,600 --> 00:22:14,720
So let me give you
the Hollywood version of that history.

479
00:22:14,720 --> 00:22:16,200
So what happened next.

480
00:22:16,200 --> 00:22:19,200
So you know this is the situation in 2013.

481
00:22:19,240 --> 00:22:21,280
You know theory is going far
ahead of experiment.

482
00:22:21,280 --> 00:22:22,200
The reason you can tell this

483
00:22:22,200 --> 00:22:23,920
in experimentalists
is they're wearing a tie

484
00:22:23,920 --> 00:22:25,920
even when they're,
you know, relaxing at home.

485
00:22:25,920 --> 00:22:27,200
So, you know, they're very distinguished.

486
00:22:29,320 --> 00:22:32,040
2013 to
2018, because they went to experiments,

487
00:22:32,040 --> 00:22:35,440
theorists were doing a lot of very, very
esoteric questions, ignoring experiment.

488
00:22:35,600 --> 00:22:38,600
In the meantime, experimentalists
were working very, very hard.

489
00:22:38,960 --> 00:22:41,640
And so to come to the natural end of that

490
00:22:41,640 --> 00:22:44,640
story, around 2018, there was new physics.

491
00:22:44,760 --> 00:22:45,080
Right.

492
00:22:45,080 --> 00:22:47,520
And I want to tell you
about that new physics and the time

493
00:22:47,520 --> 00:22:50,040
that's left
and think I have about 20 minutes left.

494
00:22:50,040 --> 00:22:52,240
So that's roughly right. Great.

495
00:22:52,240 --> 00:22:52,920
Okay.

496
00:22:52,920 --> 00:22:55,800
So the new physics is
something called the Moray effect.

497
00:22:55,800 --> 00:22:58,560
And so I went onto the internet
and looked for a figure.

498
00:22:58,560 --> 00:23:01,920
And you get this rather
nice piece of poetry which says, you know,

499
00:23:02,000 --> 00:23:05,520
which is, you know, making a riff on
an old song by Dean Martin.

500
00:23:05,800 --> 00:23:06,360
Right.

501
00:23:06,360 --> 00:23:09,360
So the Moray phenomenon
is this sort of remarkable thing.

502
00:23:09,520 --> 00:23:12,480
And, you know, it's sadly
often capital ized because it sounds like

503
00:23:12,480 --> 00:23:13,520
it's the name of a person.

504
00:23:13,520 --> 00:23:15,600
The actual story is really interesting.

505
00:23:15,600 --> 00:23:17,560
It's a word that comes from the textile
industry.

506
00:23:17,560 --> 00:23:20,840
So Moray Fabrics
actually has a long etymology.

507
00:23:20,840 --> 00:23:24,240
All the way from Arabic to English
and then via French, back to English.

508
00:23:24,600 --> 00:23:27,360
And what it is basically
is that two things.

509
00:23:27,360 --> 00:23:30,040
If I were wearing stripes
and you went back and watch this video,

510
00:23:30,040 --> 00:23:31,440
which I doubt you would do,

511
00:23:32,520 --> 00:23:34,200
you would see actually that because

512
00:23:34,200 --> 00:23:37,200
I'm being recorded on a screen
with some refresh rate.

513
00:23:37,200 --> 00:23:41,000
Often what you can see
is that when people have two screens

514
00:23:41,000 --> 00:23:43,520
or when they have striped patterns
and they're being interrogated

515
00:23:43,520 --> 00:23:47,480
at different frequency, you can see this
weird pattern of, beating interference.

516
00:23:47,760 --> 00:23:49,680
And so that's something
that often comes in

517
00:23:49,680 --> 00:23:52,680
and that's sort of a personal experience
many of us have with Moray patterns.

518
00:23:52,960 --> 00:23:54,720
So what it is,
it's really a very simple idea.

519
00:23:54,720 --> 00:23:56,440
If you have two periodic structures

520
00:23:56,440 --> 00:23:59,120
and for whatever reason,
they have slightly in one dimension,

521
00:23:59,120 --> 00:24:02,520
if they have slightly different,
frequency is or in two dimensions,

522
00:24:02,520 --> 00:24:05,520
as you'll see in a minute,
they have slightly different orientations.

523
00:24:05,640 --> 00:24:08,840
Then you can actually get a certain type
of interference pattern.

524
00:24:08,840 --> 00:24:10,320
So let's see how that works.

525
00:24:11,520 --> 00:24:13,240
So let's just take two waves.

526
00:24:13,240 --> 00:24:17,760
So I'm taking two waves which have a very
so this delta is much smaller than k.

527
00:24:17,760 --> 00:24:18,720
What I'm going to think about.

528
00:24:18,720 --> 00:24:21,360
So let's do waves
that very slightly differ in their

529
00:24:21,360 --> 00:24:23,440
you know, they're slightly offset
with respect to each other.

530
00:24:23,440 --> 00:24:26,440
And you can see that
the red and the blue and I add them up.

531
00:24:26,480 --> 00:24:27,320
And what do you see

532
00:24:27,320 --> 00:24:31,000
coming out of this is this potential
that looks awfully like it.

533
00:24:31,000 --> 00:24:32,640
Lot like it's periodic.

534
00:24:32,640 --> 00:24:33,840
It isn't really.

535
00:24:33,840 --> 00:24:35,520
It turns out that it's only periodic.

536
00:24:35,520 --> 00:24:36,680
If somehow there's some kind of

537
00:24:36,680 --> 00:24:40,640
rational relationship between delta
and K, you can think about this function

538
00:24:40,640 --> 00:24:41,800
and see if it's really periodic.

539
00:24:41,800 --> 00:24:44,920
And you can see that
unless the peaks and troughs of what

540
00:24:44,960 --> 00:24:48,000
delta wants to do exactly
line up with the peaks and troughs of K,

541
00:24:48,000 --> 00:24:52,680
this won't be really periodic, but gosh,
it looks really very close to periodic.

542
00:24:52,680 --> 00:24:54,720
So it's faking me
into thinking it's periodic.

543
00:24:54,720 --> 00:24:56,320
And so this is the moiré effect.

544
00:24:56,320 --> 00:24:59,560
When you take two things
which take two periodic functions

545
00:24:59,640 --> 00:25:02,400
and add them up,
and they have a very slightly offset

546
00:25:02,400 --> 00:25:07,040
frequency, then you get an emergent
approximate periodicity, which is much,

547
00:25:07,040 --> 00:25:10,320
much, much bigger than the microscopic
periodicity of either of them.

548
00:25:10,560 --> 00:25:10,720
Right.

549
00:25:10,720 --> 00:25:13,720
So they both started out with
a periodicity that's about two pi over k.

550
00:25:13,760 --> 00:25:15,840
But this sort of emergent long wavelength

551
00:25:15,840 --> 00:25:18,840
periodicity is something like
two pi over delta, which is really big.

552
00:25:19,120 --> 00:25:20,880
So that's an emergent length scale.

553
00:25:20,880 --> 00:25:23,440
So this is all I can do in one dimension.

554
00:25:23,440 --> 00:25:26,920
But if you're in two dimensions you can do
something a little bit more exciting.

555
00:25:26,920 --> 00:25:30,720
So let me just pass that out so
you can see for yourself how that works.

556
00:25:31,800 --> 00:25:33,240
So you can make two dimensional

557
00:25:33,240 --> 00:25:37,080
atomically thin materials
and you can actually play with those.

558
00:25:37,080 --> 00:25:40,080
So pass these up since because that side

559
00:25:41,160 --> 00:25:43,360
and you can slap layers on
if you can just pick one

560
00:25:43,360 --> 00:25:46,360
and pass them up.

561
00:25:46,880 --> 00:25:48,600
You can take two atomically thin

562
00:25:48,600 --> 00:25:51,640
layers, place them on top of each other
and nobody.

563
00:25:51,680 --> 00:25:54,720
It turns out that the kind of atomically
thin materials like graphene,

564
00:25:54,840 --> 00:25:57,720
one reason you can make them is because

565
00:25:57,720 --> 00:26:00,720
they crystallize
two very strong bonds in two dimensions.

566
00:26:00,920 --> 00:26:03,680
But the interactions between
the layers are held together

567
00:26:03,680 --> 00:26:05,160
are only Vander Walls interactions.

568
00:26:05,160 --> 00:26:06,760
They're very, very, very weak.

569
00:26:06,760 --> 00:26:09,800
So it turns out the way people
first made graphene was remarkable.

570
00:26:10,240 --> 00:26:12,560
What they did was
they took a piece of Scotch tape,

571
00:26:12,560 --> 00:26:15,160
stuck it on something that was effectively
a very, very pure

572
00:26:15,160 --> 00:26:18,360
piece of pencil lead and pulled it off
and just kept doing this.

573
00:26:18,600 --> 00:26:21,800
And if you do that enough times, you peel
off an individual atomically thin layer.

574
00:26:22,320 --> 00:26:22,680
Right.

575
00:26:22,680 --> 00:26:25,720
But you can do this twice
and by a lot of very hard work.

576
00:26:25,720 --> 00:26:29,160
And I don't want to this is exactly
what the hard graft of experimentalists.

577
00:26:29,320 --> 00:26:31,920
So my friend Andrea Young
was one of the first people who did this.

578
00:26:31,920 --> 00:26:33,360
It was mentioned in Steve Stock.

579
00:26:33,360 --> 00:26:36,720
And he roughly says, it's like you had
a kilometer long chopstick in each hand.

580
00:26:36,960 --> 00:26:38,440
You're taking a layer of clingfilm

581
00:26:38,440 --> 00:26:40,760
and trying to rotate it
another layer of clingfilm,

582
00:26:40,760 --> 00:26:43,120
all while not being able
to see what you're doing right.

583
00:26:43,120 --> 00:26:44,840
So that's roughly what people did.

584
00:26:44,840 --> 00:26:45,600
But they could do that.

585
00:26:45,600 --> 00:26:48,880
They could take two atomically thin
layers, align them close to each other,

586
00:26:48,880 --> 00:26:50,880
and then just give them
a little bit of a twist.

587
00:26:50,880 --> 00:26:53,760
And so when you do that, you get patterns
like this.

588
00:26:53,760 --> 00:26:54,000
Right.

589
00:26:54,000 --> 00:26:56,160
So the left is two kind of stripy
patterns.

590
00:26:56,160 --> 00:27:00,280
The right is what happens when you take
two hexagonal lattices and rotate them.

591
00:27:00,600 --> 00:27:02,480
And there's a little bit of algebra
you say imagine

592
00:27:02,480 --> 00:27:06,640
each thing had a periodicity
that was cosine k dot x in each direction.

593
00:27:07,320 --> 00:27:10,440
Well the angle between the two vectors,
they have the same length,

594
00:27:10,440 --> 00:27:12,400
but they're rotated
relative to each other.

595
00:27:12,400 --> 00:27:15,400
And if you think about the difference
between these two vectors

596
00:27:15,480 --> 00:27:16,440
at small angles, it's

597
00:27:16,440 --> 00:27:20,640
just the size of the vector times theta,
sort of basic trigonometric calculation.

598
00:27:21,120 --> 00:27:24,600
And so if you think about the fact
that the original periodicity

599
00:27:24,600 --> 00:27:28,040
was the lattice scale,
which is the previously advertised

600
00:27:28,040 --> 00:27:30,600
0.3 nanometer scale,
that number was cleverly chosen

601
00:27:30,600 --> 00:27:33,720
because it is roughly
the lattice spacing of, graphene.

602
00:27:34,160 --> 00:27:37,040
And if I pick an angle
of about a degree and a half,

603
00:27:37,040 --> 00:27:41,400
it turns out that this Maryland scale is
about that magic number of ten nanometers.

604
00:27:41,880 --> 00:27:44,760
So what it means is,
if I made one of these samples

605
00:27:44,760 --> 00:27:48,240
and used it as one plate of my capacitor,
that has a potential

606
00:27:48,240 --> 00:27:50,280
that's exactly match to what I can put

607
00:27:50,280 --> 00:27:54,200
push charge on and off at the one charge
per unit cell level easily.

608
00:27:54,440 --> 00:27:55,960
Right. So I can do that.

609
00:27:55,960 --> 00:27:59,520
So of course I'm not doing anything close
to putting one

610
00:27:59,520 --> 00:28:01,160
charge per microscopic cell.

611
00:28:01,160 --> 00:28:02,360
What I'm doing is pushing off

612
00:28:02,360 --> 00:28:05,360
one one electron
in every one of these yellow squares.

613
00:28:05,360 --> 00:28:07,840
I can make one of these
yellow little shaded regions.

614
00:28:07,840 --> 00:28:09,640
I can pop an electron on and off.

615
00:28:09,640 --> 00:28:11,840
So it's a very,
very low density of electrons.

616
00:28:11,840 --> 00:28:14,760
And normally there wouldn't be a potential
for it to push against.

617
00:28:14,760 --> 00:28:17,840
But by twisting this lattice
I've created a new potential.

618
00:28:17,840 --> 00:28:20,600
It's a much, much lower energy scale
than the original potential.

619
00:28:20,600 --> 00:28:24,120
So while the original answer
would be a long run electron volt

620
00:28:24,120 --> 00:28:25,880
or a 10th of an electron volt scale,

621
00:28:25,880 --> 00:28:29,560
we're down now to the electron volt scale
because a much, much lower energy scale.

622
00:28:29,800 --> 00:28:32,800
But if I do a very sensitive experiment,
I can measure things of that scale

623
00:28:32,800 --> 00:28:34,440
so that I'm now going to be playing

624
00:28:34,440 --> 00:28:37,440
with a periodic potential
that sits at that scale.

625
00:28:37,440 --> 00:28:40,680
And another fact is that
just as the magnetic field modifies

626
00:28:40,680 --> 00:28:44,080
the kinetic energy of free electrons,
the fact that you have this potential

627
00:28:44,280 --> 00:28:47,640
turns out to give you flat bands
with a very small kinetic energy.

628
00:28:48,000 --> 00:28:49,280
And the great thing about this is,

629
00:28:49,280 --> 00:28:52,520
and I'll come back to this
in a little bit, there's a huge number of,

630
00:28:53,560 --> 00:28:54,960
kinds of two dimensional materials.

631
00:28:54,960 --> 00:28:58,160
Graphene was the first discovered, but
they're now a huge the libraries of them.

632
00:28:58,160 --> 00:29:00,160
I'll come back to that.
So it means that there's essentially

633
00:29:00,160 --> 00:29:03,160
combinatorial possibilities
for new physics.

634
00:29:03,240 --> 00:29:06,480
But the hydrogen atom of that new field
is basically taking

635
00:29:06,480 --> 00:29:08,320
two graphene layers and twisting them.

636
00:29:08,320 --> 00:29:12,520
And so there's some extra details because
graphene famously does not have electrons

637
00:29:12,520 --> 00:29:15,600
that look like free particles,
they look more like Dirac particles.

638
00:29:15,800 --> 00:29:17,520
And that actually gives you
some extra richness,

639
00:29:17,520 --> 00:29:21,520
which is noted by these two gentlemen,
Raffi Spritzer and Alan McDonald.

640
00:29:21,760 --> 00:29:24,600
And they actually came up with a way,
it turns out, to actually figure out

641
00:29:24,600 --> 00:29:26,160
what this lattice does.

642
00:29:26,160 --> 00:29:28,840
This new emergent
super lattice does is not easy.

643
00:29:28,840 --> 00:29:30,880
You have to figure out new ways
to calculate with them.

644
00:29:30,880 --> 00:29:31,840
And they did that.

645
00:29:31,840 --> 00:29:35,160
And what you can I hope what you can see
here is by playing around with the angles.

646
00:29:35,360 --> 00:29:38,200
You know that one year, that 1.25 degrees,

647
00:29:38,200 --> 00:29:41,040
what you see is a very,
very flat band there.

648
00:29:41,040 --> 00:29:42,040
So technical reasons

649
00:29:42,040 --> 00:29:43,800
why they're actually eight bands in there,

650
00:29:43,800 --> 00:29:45,600
because electrons
come with different spins

651
00:29:45,600 --> 00:29:47,480
and they have other
kind of labels on them.

652
00:29:47,480 --> 00:29:48,960
But you get these eight energy bands.

653
00:29:48,960 --> 00:29:52,560
And so from what I've told you so far,
what you should be thinking about

654
00:29:52,560 --> 00:29:55,880
is that, well,
I've got some boring insulator here.

655
00:29:55,880 --> 00:29:59,400
Relatively boring, because after all,
I just filled up these bands.

656
00:29:59,400 --> 00:30:01,440
It looks like the insulator in my textbook

657
00:30:01,440 --> 00:30:04,440
because I just have filled bands
and then a gap to the next band.

658
00:30:04,480 --> 00:30:06,360
I should get a boring insulator here.

659
00:30:06,360 --> 00:30:09,440
And if you count
those boring insulators happen

660
00:30:10,200 --> 00:30:12,480
at fillings of minus four.

661
00:30:12,480 --> 00:30:14,960
And for why there fillings of minus
four and four

662
00:30:14,960 --> 00:30:17,880
is that graphene
has some natural electron count, so minus

663
00:30:17,880 --> 00:30:21,320
four means I pull away four electrons
in every one of those cells.

664
00:30:21,320 --> 00:30:23,800
Plus four means I add four electrons
to each of those cells

665
00:30:23,800 --> 00:30:26,920
so I can look at things adding
and subtracting charge to that system.

666
00:30:27,560 --> 00:30:31,520
But if I'm inside there,
I have a really interesting problem

667
00:30:31,520 --> 00:30:32,760
because essentially the kinetic.

668
00:30:32,760 --> 00:30:36,640
So this the kind of the width of this
band, the thickness and energy

669
00:30:36,680 --> 00:30:38,880
gives me a figure of metric
for the kinetic energy.

670
00:30:38,880 --> 00:30:41,880
And actually the Coulomb interactions
are much, much bigger than this.

671
00:30:42,240 --> 00:30:45,240
So it means that the electrons here
are very, very strongly correlated.

672
00:30:45,320 --> 00:30:47,080
Which means if I put charge on there,

673
00:30:47,080 --> 00:30:48,720
anything I see should be coming out

674
00:30:48,720 --> 00:30:50,760
as the result of the electrons
talking to each other,

675
00:30:50,760 --> 00:30:52,720
rather than they're
talking to the super lattice.

676
00:30:54,320 --> 00:30:55,080
And so the

677
00:30:55,080 --> 00:30:58,760
group of Pablo Hurler who wrote
MIT did this in about 2018.

678
00:30:59,000 --> 00:31:02,680
And in fact, what they found
was that, yes, you see, the boring

679
00:31:02,680 --> 00:31:06,520
and B ice for band insulator may as well
be for boring insulator for this thing.

680
00:31:06,520 --> 00:31:08,480
So there's nothing
interesting happening there.

681
00:31:08,480 --> 00:31:10,280
But what they saw
was something very surprising.

682
00:31:10,280 --> 00:31:11,400
So you notice that there two.

683
00:31:11,400 --> 00:31:14,600
So what this curve is, is just showing you
how conducting the sample is

684
00:31:14,600 --> 00:31:18,240
as I tune the gate voltage, as I tune
push charge on and off the system.

685
00:31:18,480 --> 00:31:19,800
And what you see is while it's

686
00:31:19,800 --> 00:31:22,720
really an insulator here and here
that I would have expected, that just came

687
00:31:22,720 --> 00:31:26,720
from this calculation of electrons
talking to the weird moiré lattice.

688
00:31:27,240 --> 00:31:30,760
But what happens here are two
new insulators that nobody expected.

689
00:31:31,200 --> 00:31:32,240
Why is that insulating?

690
00:31:32,240 --> 00:31:35,120
It must be as a result of electron
electron interactions.

691
00:31:35,120 --> 00:31:35,640
So it turns out

692
00:31:35,640 --> 00:31:38,760
there are about 100 different proposals
for what that insulator was.

693
00:31:38,760 --> 00:31:40,880
And sort of by luck or good judgment,

694
00:31:40,880 --> 00:31:42,840
Steve
and I managed to propose the correct one.

695
00:31:42,840 --> 00:31:45,800
And that was confirmed in experiments
a year or so ago.

696
00:31:45,800 --> 00:31:47,760
And there's another surprise,
which is a surprise

697
00:31:47,760 --> 00:31:49,080
that still remains to be explained.

698
00:31:49,080 --> 00:31:52,080
So if you take those insulators
and move a little bit away from them,

699
00:31:52,080 --> 00:31:53,320
you get a superconductor.

700
00:31:53,320 --> 00:31:54,360
Nobody expected that.

701
00:31:54,360 --> 00:31:57,360
Nobody expected graphene to become
a superconductor in this situation.

702
00:31:57,600 --> 00:31:59,880
So you've taken an insulator,
you've added a little bit of charge,

703
00:31:59,880 --> 00:32:02,880
and somehow, magically,
the added charge becomes superconducting.

704
00:32:03,000 --> 00:32:04,760
We don't know why
it becomes superconducting.

705
00:32:04,760 --> 00:32:07,080
There's a huge debate in the literature
over this.

706
00:32:07,080 --> 00:32:08,760
We don't know what that superconductor is,

707
00:32:08,760 --> 00:32:12,440
whether it's like the old superconductors
that people discovered in the 19th.

708
00:32:12,480 --> 00:32:15,200
You know, that people explain
successfully in the 1950s

709
00:32:15,200 --> 00:32:16,920
or the new high temperature
superconductors.

710
00:32:16,920 --> 00:32:18,360
We don't know what the mechanisms are.

711
00:32:18,360 --> 00:32:21,040
This is a big area of research that
lots of people are trying to understand,

712
00:32:22,920 --> 00:32:26,280
but and there's a whole bunch of other
phenomena and some hints of topology.

713
00:32:26,480 --> 00:32:30,120
But it turned out that in graphene
it was very, very, very hard to get these

714
00:32:30,120 --> 00:32:31,800
topological states to reliably.

715
00:32:31,800 --> 00:32:35,560
And in the end, the only way to get them
reliably was to go back to the old idea

716
00:32:35,560 --> 00:32:38,960
of turning on a little bit of a turning
on a magnetic field and teasing them out.

717
00:32:39,240 --> 00:32:42,080
So that wasn't very satisfactory,
because after all, the way we went into

718
00:32:42,080 --> 00:32:45,120
this was getting rid of the magnetic field
entirely and trying to do this

719
00:32:45,120 --> 00:32:48,120
without a big magnetic field,
without all these exotic effects.

720
00:32:48,440 --> 00:32:51,120
So we have a nice system,

721
00:32:51,120 --> 00:32:54,120
but only tantalizing hints of topology,
and so can we do better.

722
00:32:54,880 --> 00:32:57,400
So indeed, the answer is that we can.

723
00:32:57,400 --> 00:33:00,000
And the reason that we can do better
is because of the fact

724
00:33:00,000 --> 00:33:03,000
that there's this combinatorial richness
to this new class of platforms,

725
00:33:03,120 --> 00:33:04,400
you know, old material science.

726
00:33:04,400 --> 00:33:07,520
You needed to know a lot of chemistry
because you needed to know, you know, what

727
00:33:07,520 --> 00:33:10,680
things will combine with other things
to form, you know, good compounds.

728
00:33:10,680 --> 00:33:12,480
How would those compounds crystallize?

729
00:33:12,480 --> 00:33:15,760
Here are the nice idea
here is that you have these 2D materials

730
00:33:15,920 --> 00:33:18,000
and you can play combinatorics
instead of chemistry.

731
00:33:18,000 --> 00:33:19,920
You can stack the materials
on top of each other.

732
00:33:19,920 --> 00:33:21,960
You can twist them.
You can pick different materials.

733
00:33:21,960 --> 00:33:26,160
You can try and use different place
the resulting thing on a different

734
00:33:26,440 --> 00:33:29,600
substrate to change the nature
of interactions between electrons.

735
00:33:30,000 --> 00:33:32,960
And so if you look at the library of 2D

736
00:33:32,960 --> 00:33:35,960
materials, there's actually a lot of them
that can be Scotch taped.

737
00:33:36,000 --> 00:33:39,000
Of course, to get the best samples, you do
something different than scotch tape.

738
00:33:39,000 --> 00:33:41,480
You do chemical vapor deposition
or something like that.

739
00:33:41,480 --> 00:33:43,840
But still,
the Scotch tape idea is pretty cool.

740
00:33:43,840 --> 00:33:45,480
And so there's a whole list of them
right there.

741
00:33:45,480 --> 00:33:48,640
And so I'm not
going to go into any of them except one.

742
00:33:48,800 --> 00:33:49,680
So there's a very important

743
00:33:49,680 --> 00:33:53,160
class of things called transition
metal DHL coordinates tm, MDS.

744
00:33:53,480 --> 00:33:56,040
There's molybdenum disulfide,
which I think for many years

745
00:33:56,040 --> 00:33:57,840
is used as a lubricant
or something like that.

746
00:33:57,840 --> 00:34:00,960
It turns out to be a very important
compound for a lot of these experiments.

747
00:34:01,120 --> 00:34:03,680
And there's a whole family
and I'm going to focus on these.

748
00:34:03,680 --> 00:34:06,880
And so what these are it doesn't
look like this in this rendering.

749
00:34:06,880 --> 00:34:08,520
But they also have a honeycomb lattice.

750
00:34:08,520 --> 00:34:10,080
That's it looks a little bit different
than graphene.

751
00:34:10,080 --> 00:34:13,080
But if I looked at it on the 2D plane
it would have some honeycomb like pattern.

752
00:34:13,440 --> 00:34:15,720
And it's made up of the unpacking
the name.

753
00:34:15,720 --> 00:34:16,920
It's made up of transition

754
00:34:16,920 --> 00:34:20,160
metals like tungsten,
like molybdenum, tungsten or hafnium.

755
00:34:20,760 --> 00:34:21,600
And a child.

756
00:34:21,600 --> 00:34:23,960
A Charcot gene is a fancy word
for something in the same row

757
00:34:23,960 --> 00:34:25,240
as sulfur in the periodic table.

758
00:34:25,240 --> 00:34:27,360
So sulfur selenium or tellurium.

759
00:34:27,360 --> 00:34:32,160
So the characterized by this rather neat
fact that in any individual layer

760
00:34:32,320 --> 00:34:35,280
they just have they have something
that looks like a parabolic dispersion.

761
00:34:35,280 --> 00:34:38,080
You don't have to worry too much.
And so it looks like we did all of this.

762
00:34:38,080 --> 00:34:41,160
It's like it's not as rich as graphene,
an individual layer,

763
00:34:41,480 --> 00:34:44,880
because graphene, after all, had this
fancy Dirac electron floating around.

764
00:34:44,880 --> 00:34:46,240
None of that happens here.

765
00:34:46,240 --> 00:34:48,360
Instead you have this just

766
00:34:48,360 --> 00:34:52,040
it looks as though a apart from some label
that the mass is a little bit different.

767
00:34:52,240 --> 00:34:53,280
It looks as though you could just

768
00:34:53,280 --> 00:34:55,920
start off with ordinary electrons
and free space.

769
00:34:55,920 --> 00:34:59,760
But what's nice about it is that
if you look, if you now take one layer

770
00:34:59,760 --> 00:35:03,240
of this TMD and place it another layer,
now electrons want to tumble

771
00:35:03,240 --> 00:35:07,440
between the layers and the potential
that the two electrons, that electrons

772
00:35:07,440 --> 00:35:10,920
moving in this combined system
see are because of that tunneling.

773
00:35:11,200 --> 00:35:13,000
But that tunneling
takes a particular form.

774
00:35:13,000 --> 00:35:16,000
So this delta captures
the tunneling between the layers.

775
00:35:16,240 --> 00:35:19,480
And that essentially all that
tunneling ends up doing is

776
00:35:19,680 --> 00:35:21,360
there's also a different potential
in the layer.

777
00:35:21,360 --> 00:35:25,000
So that delta
so sigma is some Pauli matrix.

778
00:35:25,000 --> 00:35:26,240
I won't write it down that.

779
00:35:26,240 --> 00:35:29,320
Just think of it as, you know,
just like I have spin.

780
00:35:29,520 --> 00:35:32,200
There's ups and downs,
but I can represent it as a spin.

781
00:35:32,200 --> 00:35:36,120
Imagine Sigma Plus is in the top layer,

782
00:35:36,400 --> 00:35:38,600
Sigma pointing down is in the bottom
layer.

783
00:35:38,600 --> 00:35:42,480
And Sigma in the plane would be you're in
one, you're in some superposition

784
00:35:42,480 --> 00:35:44,520
of the two layers
with some definite phase.

785
00:35:44,520 --> 00:35:46,360
So it turns out that

786
00:35:47,440 --> 00:35:50,440
this one second.

787
00:35:52,880 --> 00:35:54,480
What you find

788
00:35:54,480 --> 00:35:58,080
is that this delta is like a period.

789
00:35:58,080 --> 00:35:59,160
It's a periodic field.

790
00:35:59,160 --> 00:36:02,160
It's got the periodicity
of this emergent moiré supercell.

791
00:36:02,280 --> 00:36:05,600
And that's like a Zeeman field that
tells you which layer you want to be in.

792
00:36:05,880 --> 00:36:08,040
So the electron really wants to be in
the layer.

793
00:36:08,040 --> 00:36:09,600
It's told to be by that.

794
00:36:09,600 --> 00:36:11,920
And the kinetic energy
goes along for the ride

795
00:36:11,920 --> 00:36:13,560
because it doesn't care
which layer you're in.

796
00:36:13,560 --> 00:36:15,000
Roughly speaking.

797
00:36:15,000 --> 00:36:18,680
And it turns out that near
some special angles, the

798
00:36:19,360 --> 00:36:22,520
this delta forms a lattice,
which is known as a skirmish on lattice.

799
00:36:22,520 --> 00:36:24,360
So Shivaji sketch what a skirmish on was.

800
00:36:24,360 --> 00:36:27,760
It's this configuration
where, where you kind of wrap the sphere.

801
00:36:28,040 --> 00:36:31,000
So this is a kind of busy plot,
but what this is saying is

802
00:36:31,000 --> 00:36:34,560
where this thing is yellow, roughly
speaking, and the color is dark.

803
00:36:35,000 --> 00:36:38,080
Your electron is entirely in the bottom
layer where it's white.

804
00:36:38,080 --> 00:36:39,680
You are entirely in the top layer.

805
00:36:39,680 --> 00:36:41,680
And where it's in between,
you have these blue arrows

806
00:36:41,680 --> 00:36:44,840
that tell you where on the equator
you point and if you look at this

807
00:36:45,360 --> 00:36:48,000
little diamond here, that's one unit cell.

808
00:36:48,000 --> 00:36:50,960
And if you count the topological index
in that unit cell,

809
00:36:50,960 --> 00:36:54,560
following the rules that Shivaji gave you,
there's one skirmish in that unit cell.

810
00:36:54,760 --> 00:36:56,960
You go to the next unit
cell, there's another skirmish on.

811
00:36:56,960 --> 00:36:58,560
Now the skirmish on
is in an abstract space.

812
00:36:58,560 --> 00:37:01,560
It doesn't really spin. It's
this layer on.

813
00:37:01,560 --> 00:37:04,960
But it turns out that this has really
interesting consequences for the electrons

814
00:37:05,400 --> 00:37:08,040
to see that
what you do is do something that we're

815
00:37:08,040 --> 00:37:11,520
all taught to do is always go to a frame
in which the problem looks simple.

816
00:37:11,760 --> 00:37:15,480
So we rotate so that we're always pointing
in the our layer

817
00:37:15,480 --> 00:37:19,080
coordinate is always pointing along
the way we're told to point by delta.

818
00:37:19,560 --> 00:37:23,320
And if you do that change of basis,
what you find is that the problem reduces

819
00:37:23,320 --> 00:37:26,640
to looking exactly like
you get for electrons in a magnetic field,

820
00:37:26,720 --> 00:37:29,800
like there's a vector potential
that electrons see, but there's also some

821
00:37:29,800 --> 00:37:32,800
effective potential landscape
that they see.

822
00:37:32,960 --> 00:37:35,960
And each of these things is periodic
with the periodicity of the unit cell.

823
00:37:36,400 --> 00:37:40,880
And it turns out that
if this has once coming on per unit cell,

824
00:37:41,040 --> 00:37:42,560
that corresponds almost exactly, it

825
00:37:42,560 --> 00:37:46,200
corresponds exactly to having one flux
quantum per unit cell in this new problem.

826
00:37:46,520 --> 00:37:49,320
So what you've just done is realize,

827
00:37:49,320 --> 00:37:52,040
effectively a churn band,
which is pretty flat.

828
00:37:52,040 --> 00:37:55,240
It turns out that if you the dust
settles, that's what you get. So

829
00:37:56,400 --> 00:37:59,480
let's go back to our wish list
and see whether we get the thing.

830
00:37:59,480 --> 00:38:01,560
So we have
it turns out there's a particular example

831
00:38:01,560 --> 00:38:05,280
that's very nice
where the transition metal is molybdenum.

832
00:38:05,280 --> 00:38:06,520
The gene is tellurium.

833
00:38:06,520 --> 00:38:11,160
So molybdenum Telluride
and you know, in typical physics fashion,

834
00:38:11,680 --> 00:38:13,560
when you twist them,
you put a little T in front.

835
00:38:13,560 --> 00:38:16,560
So it's tmo
t two is the fancy word for this material.

836
00:38:16,760 --> 00:38:20,600
And it turns out if you take the angle
of about 3.7 degrees, you get a churn band

837
00:38:20,640 --> 00:38:22,480
that has a good gap
to all the other bands.

838
00:38:22,480 --> 00:38:25,480
So you can get this topology,
you've got the topology in there.

839
00:38:25,760 --> 00:38:27,560
The bands are flat
with the kinetic energy.

840
00:38:27,560 --> 00:38:29,600
That's much less than the interactions.

841
00:38:29,600 --> 00:38:32,720
So you've got correlations
and there's tune ability.

842
00:38:32,720 --> 00:38:35,720
So the lattice
spacing of the scale is about comparable.

843
00:38:35,760 --> 00:38:38,760
It gives you a length
of about ten nanometers which is exactly

844
00:38:38,760 --> 00:38:41,760
what we can dial on and off
with a one volt potential.

845
00:38:42,120 --> 00:38:45,160
And you can also
it turns out that the interlayer potential

846
00:38:45,240 --> 00:38:48,240
can be used to switch on
and off the topology,

847
00:38:48,240 --> 00:38:50,680
which is kind of nice
because sometimes it's nice.

848
00:38:50,680 --> 00:38:52,320
One of the best ways to show
that you've got

849
00:38:52,320 --> 00:38:54,640
something is to show that you can control.
Turn it off.

850
00:38:54,640 --> 00:38:55,560
And if you're an experimentalist,

851
00:38:55,560 --> 00:38:58,400
you really like the ability
to switch things on and off.

852
00:38:58,400 --> 00:39:00,480
And in fact, when people made the sample.

853
00:39:00,480 --> 00:39:03,320
So this is actually the second experiment
that were done on this.

854
00:39:03,320 --> 00:39:05,640
Now they'll come back
to the first experiments in a minute.

855
00:39:05,640 --> 00:39:06,520
So what you should see

856
00:39:06,520 --> 00:39:10,600
here is on the x axis of these plots
is that interlayer displacement field.

857
00:39:10,920 --> 00:39:14,120
When that zero the system forms
the on lattice.

858
00:39:14,320 --> 00:39:16,680
And the blue curve
is the whole resistance.

859
00:39:16,680 --> 00:39:18,960
And the red curve
is the longitudinal resistance.

860
00:39:18,960 --> 00:39:22,800
So you see the telltale signature
of a quantum Hall like response, a zero

861
00:39:22,800 --> 00:39:26,320
longitudinal resistance and a quantized
hold resistance at h over E squared.

862
00:39:26,520 --> 00:39:29,120
So you just created
an integer Chern insulator.

863
00:39:29,120 --> 00:39:32,000
But if you tweak the displacement field
above a critical value,

864
00:39:32,000 --> 00:39:35,880
you turn off this scummy
on lattice and bang, everything goes away.

865
00:39:36,320 --> 00:39:38,960
So you've actually controllable,
created a quantum state.

866
00:39:38,960 --> 00:39:41,680
You couldn't
you couldn't do this in an a lambda level

867
00:39:41,680 --> 00:39:44,680
because you don't have a knob
to switch on and off the topology.

868
00:39:45,040 --> 00:39:49,080
Even more remarkably, if you now,
you know, work a little bit harder,

869
00:39:49,080 --> 00:39:50,960
you can actually focus
on a different density.

870
00:39:50,960 --> 00:39:53,600
So going from here
to here is tuning the other knob,

871
00:39:53,600 --> 00:39:56,600
turning a dial on my gate
so I can change the charge

872
00:39:56,880 --> 00:40:00,480
and at this point you're
partially filling this band and you find

873
00:40:00,480 --> 00:40:04,120
that there's a whole resistance of,
three over two e squared.

874
00:40:04,400 --> 00:40:07,080
Or if you translate that
into conductivity, that's a two thirds.

875
00:40:07,080 --> 00:40:09,200
That's like a two thirds
quantum Hall state.

876
00:40:09,200 --> 00:40:11,760
There's an active search
for using these platforms

877
00:40:11,760 --> 00:40:14,520
to get a whole list of,
you know, exotic states.

878
00:40:14,520 --> 00:40:17,640
So non-abelian
anyons, like Steve talked about something

879
00:40:17,840 --> 00:40:20,840
more exotic states
called fractional topological insulators.

880
00:40:21,160 --> 00:40:25,240
But again, one of the really remarkable
things of why progress in this field

881
00:40:25,240 --> 00:40:28,200
is exciting is that, you know,
if you go back to the quantum Hall effect.

882
00:40:28,200 --> 00:40:31,120
So Steve talked about the work of Swiss
armor and Garside,

883
00:40:31,120 --> 00:40:33,320
who are working in gallium
arsenide samples.

884
00:40:33,320 --> 00:40:35,480
So for the fraction of quantum
Hall effect,

885
00:40:35,480 --> 00:40:38,560
you know, other groups replicated it
in gallium arsenide, but for it

886
00:40:38,560 --> 00:40:41,720
to be replicated
in another material took about ten years.

887
00:40:41,720 --> 00:40:45,080
So it's a very long time for it
to be reproduced in a second material.

888
00:40:45,320 --> 00:40:46,680
So you might have been forgiven
for thinking

889
00:40:46,680 --> 00:40:49,320
maybe there's some very special feature
to this one material.

890
00:40:49,320 --> 00:40:51,440
Why should we care?
Why is it a universal phenomenon?

891
00:40:51,440 --> 00:40:53,360
Whatever a theorist tells me,

892
00:40:53,360 --> 00:40:57,760
the timescale for the fractional Chern
insulator to be reproduced was ten weeks,

893
00:40:58,480 --> 00:41:02,560
because roughly
so just a few months twisted, maybe two

894
00:41:02,880 --> 00:41:04,800
people did something
in a completely different system,

895
00:41:04,800 --> 00:41:06,240
which is taking, you know,

896
00:41:06,240 --> 00:41:09,240
when you do this exfoliation of graphene,
you're trying to get a single layer,

897
00:41:09,320 --> 00:41:12,240
but sometimes you peel off
five layers at a time.

898
00:41:12,240 --> 00:41:13,880
And somebody said,
why don't we just stick that

899
00:41:13,880 --> 00:41:16,800
five layer sample and play with it
and see what it does?

900
00:41:16,800 --> 00:41:18,960
And it turns out
you can see a whole bunch of fractions

901
00:41:18,960 --> 00:41:20,440
for the same type of experiment.

902
00:41:20,440 --> 00:41:23,800
It turns out that this physics, you know,
it's about a year and a half old.

903
00:41:23,800 --> 00:41:25,720
There's still active debate over
what's going on.

904
00:41:25,720 --> 00:41:28,080
Nobody understands
this. Do any great detail.

905
00:41:28,080 --> 00:41:30,600
Very happy to say that one of my former
students is one of the leading.

906
00:41:30,600 --> 00:41:31,840
I'm not involved in this works.

907
00:41:31,840 --> 00:41:35,120
I can say it is one of the leading people
in trying to piece together

908
00:41:35,120 --> 00:41:36,360
what's happening in these materials.

909
00:41:36,360 --> 00:41:38,880
So turns out there
could be some proposals.

910
00:41:38,880 --> 00:41:42,600
It realizes some new states of matter
that are completely different in some ways

911
00:41:42,600 --> 00:41:44,960
from what we had expected from before.

912
00:41:45,920 --> 00:41:47,640
So just
to comment a little bit on the future.

913
00:41:47,640 --> 00:41:49,200
So, you know, now that the

914
00:41:49,200 --> 00:41:52,800
tortoise of experiment has led lab us
and they're going forward,

915
00:41:53,000 --> 00:41:55,360
theorists are having to come back
and think a little bit more

916
00:41:55,360 --> 00:41:58,200
about how to model these states
and also how to look for them.

917
00:41:58,200 --> 00:42:00,360
And so one of the things
that Steve talked about

918
00:42:00,360 --> 00:42:01,880
is that a signature
of the fractional quantum

919
00:42:01,880 --> 00:42:04,800
Hall effect
is this idea of any statistics.

920
00:42:04,800 --> 00:42:09,240
And the way you see it in a Landau level
is by doing this edge state interferometry

921
00:42:09,240 --> 00:42:12,480
experiment that Shivaji proposed
and Steve has talked about today,

922
00:42:13,240 --> 00:42:16,240
it turns out there's a wrinkle of that
in these two dimensional materials.

923
00:42:16,280 --> 00:42:17,160
The wrinkle is that,

924
00:42:17,160 --> 00:42:21,120
you know, when you want to do transport
measurements, you have to take gold

925
00:42:21,120 --> 00:42:21,480
or something

926
00:42:21,480 --> 00:42:24,600
like that, or platinum and make electrodes
and stick them on the sample.

927
00:42:24,880 --> 00:42:26,360
And chemistry doesn't like that.

928
00:42:26,360 --> 00:42:29,520
It turns out that only a couple of groups,
I think at this stage, one group

929
00:42:29,520 --> 00:42:33,200
in the world knows how
to make good contacts to these MDS.

930
00:42:33,480 --> 00:42:34,520
So what other what?

931
00:42:34,520 --> 00:42:36,320
But the nice thing
about a two dimensional system is

932
00:42:36,320 --> 00:42:38,400
you can look at it
and you can look at it slightly

933
00:42:38,400 --> 00:42:40,320
more scientifically
and put light in there.

934
00:42:40,320 --> 00:42:43,040
So it turns out the first way
people looked at it is by some clever

935
00:42:43,040 --> 00:42:45,600
photo luminescence experiments
that were allowed.

936
00:42:45,600 --> 00:42:48,600
There's a signature in this
which I won't go into that explains that,

937
00:42:49,520 --> 00:42:52,680
you know, if you see a sharp
if you see a kind of kink in this photo

938
00:42:52,680 --> 00:42:54,480
luminescence, you should imagine that

939
00:42:54,480 --> 00:42:57,880
there is a fractional
there's an insulating state there.

940
00:42:58,320 --> 00:43:02,640
So this got us thinking could we actually
just use light to search for ions?

941
00:43:02,640 --> 00:43:05,160
Can we just shine light on a sample
and look for that?

942
00:43:05,160 --> 00:43:08,880
You know, one of the things we're actively
working on is can you actually measure

943
00:43:08,880 --> 00:43:11,000
fractional statistics
by shining light on samples?

944
00:43:11,000 --> 00:43:12,600
And at least in theory, the answer is yes.

945
00:43:12,600 --> 00:43:15,360
So we're working with experimental groups
to realize those ideas.

946
00:43:15,360 --> 00:43:17,800
There's just to illustrate
that having a new platform

947
00:43:17,800 --> 00:43:20,240
not only allows us to realize
all theoretical ideas,

948
00:43:20,240 --> 00:43:22,080
but forces us to think of new ways

949
00:43:22,080 --> 00:43:24,960
of doing things that we thought
we knew how to do well before.

950
00:43:24,960 --> 00:43:27,480
So just in summary, since I'm out of time,
you know,

951
00:43:27,480 --> 00:43:31,800
by the simple idea of twisting
two dimensional materials,

952
00:43:32,080 --> 00:43:33,120
we're able to

953
00:43:33,120 --> 00:43:37,000
get a confluence of three key new routes
to physics, of routes, to new physics.

954
00:43:37,200 --> 00:43:40,920
The idea that you want electronic states
with some topology,

955
00:43:41,160 --> 00:43:43,200
where electrons strongly interact
with each other,

956
00:43:43,200 --> 00:43:46,280
and where you have the ability
to actually see any of those phenomena.

957
00:43:48,480 --> 00:43:53,120
And I
think the last time and just say, and,

958
00:43:53,240 --> 00:43:56,440
you know, I just gave a particular example
of this richness of the setting

959
00:43:56,440 --> 00:43:59,200
to look for these integer
and fractional Chern insulating states.

960
00:43:59,200 --> 00:44:00,600
But there's a whole bunch of other things

961
00:44:00,600 --> 00:44:03,600
you can look for, like the superconductors
I talked about and so on.

962
00:44:04,200 --> 00:44:08,400
And so I think just to summarize this idea
of probing both familiar states,

963
00:44:08,400 --> 00:44:11,400
but also seeking new ones,
using the versatility of these platforms

964
00:44:11,560 --> 00:44:15,240
is really increasingly a center
of activity in the field of quantum

965
00:44:15,360 --> 00:44:16,440
condensed matter physics. Today.

966
00:44:16,440 --> 00:44:19,440
So let me close with that
and take your questions.

967
00:44:33,240 --> 00:44:35,760
So those are both
extremely good questions.

968
00:44:35,760 --> 00:44:36,720
So let me answer the first.

969
00:44:36,720 --> 00:44:38,400
So non-abelian anyons.

970
00:44:38,400 --> 00:44:41,320
It's not the dimension of space.
It's an interesting point.

971
00:44:41,320 --> 00:44:43,560
So Stephen
I didn't quite mention non-abelian.

972
00:44:43,560 --> 00:44:44,760
So Steve talked about anyons.

973
00:44:44,760 --> 00:44:47,160
So anyons are generalizations
of statistics

974
00:44:47,160 --> 00:44:50,560
where, you know, instead of getting a
minus one when you take a particle around,

975
00:44:50,720 --> 00:44:52,960
when you exchange two particles,
you get a phase.

976
00:44:52,960 --> 00:44:56,360
So it turns out that a class of systems
where if you take a system of,

977
00:44:56,600 --> 00:44:59,920
you know, frozen anyons
and you take one around the you swap them

978
00:45:00,160 --> 00:45:02,520
instead of getting a phase,
you get a matrix.

979
00:45:02,520 --> 00:45:05,720
And that turns out to be intimately linked
to the fact that when you have,

980
00:45:06,440 --> 00:45:08,720
essentially when you have
a bunch of anyons in your sample,

981
00:45:08,720 --> 00:45:11,840
if they're non-abelian, there are many
possible degenerate ground states.

982
00:45:11,840 --> 00:45:13,640
And when you do this transformation,
you're doing

983
00:45:13,640 --> 00:45:16,520
some kind of adiabatic rotation
between the space of those states.

984
00:45:16,520 --> 00:45:17,280
And in fact,

985
00:45:17,280 --> 00:45:21,040
when I came up with this idea of quantum
memory, really the non-abelian case

986
00:45:21,040 --> 00:45:24,280
is the one that's very rich for doing
lots of quantum protected operations.

987
00:45:24,480 --> 00:45:27,120
So that's why the non-abelian one
has this totemic significance.

988
00:45:27,120 --> 00:45:30,000
If you can achieve that, you can do a lot.
So that's the first answer.

989
00:45:30,000 --> 00:45:32,920
The second answer is really very,
very insightful because in fact,

990
00:45:32,920 --> 00:45:35,800
if you take graphene
and try and strain it,

991
00:45:35,800 --> 00:45:38,880
there are things that look a bit
like gravitational fields and so on.

992
00:45:39,120 --> 00:45:41,440
It turns out for graphene,
what really happens is that effectively

993
00:45:41,440 --> 00:45:43,840
you put on a magnetic vector
potential on the sample.

994
00:45:43,840 --> 00:45:46,720
So it's sort of related
but not quite the same thing.

995
00:45:46,720 --> 00:45:48,000
Now for more graphene.

996
00:45:48,000 --> 00:45:51,120
It turns out that in fact,
when you take two layers and stick them on

997
00:45:51,120 --> 00:45:54,240
top of each other, it may not surprise you
that, you know, while things are

998
00:45:54,520 --> 00:45:57,800
sort of happy being twisted,
they're not thrilled with the idea.

999
00:45:58,400 --> 00:45:58,960
Right?

1000
00:45:58,960 --> 00:46:00,920
And so that puts
lots of strains on the sample.

1001
00:46:00,920 --> 00:46:04,240
So there's almost never a sample that,
you know, lack some kind of strain.

1002
00:46:04,560 --> 00:46:07,440
And so, you know, most people just thought
of this as an irrelevant complication.

1003
00:46:07,440 --> 00:46:09,640
And they solved these problems
which didn't have strain in them.

1004
00:46:09,640 --> 00:46:12,840
They got all these disparate answers
for why these insulators were forming.

1005
00:46:12,840 --> 00:46:14,120
And that graphene thing,

1006
00:46:14,120 --> 00:46:18,480
I mean, led by our students and postdocs
who insisted that this was important.

1007
00:46:18,480 --> 00:46:20,880
Steve and I and our group
took that quite seriously.

1008
00:46:20,880 --> 00:46:23,640
And it turned out once
you took that seriously, that I explained

1009
00:46:23,640 --> 00:46:25,840
why there were insulators forming.
And that's now

1010
00:46:25,840 --> 00:46:29,040
the accepted explanation that it plays
a very important role indeed.

1011
00:46:29,200 --> 00:46:30,400
So it does play a role.

1012
00:46:30,400 --> 00:46:33,400
And there are ways to engineer
kind of gravitational like phenomena,

1013
00:46:33,400 --> 00:46:35,200
but not quite so straightforwardly
as that.

1014
00:46:35,200 --> 00:46:39,120
It's much easier to engineer electromagnetic phenomena with strain, if you like.

1015
00:46:40,200 --> 00:46:42,120
I don't know if there's any,

1016
00:46:42,120 --> 00:46:44,360
thing for neuroscience,
because at some level, these are,

1017
00:46:44,360 --> 00:46:47,360
you know, the kind of things
that have most natural connections

1018
00:46:47,360 --> 00:46:49,680
between sort of neuroscience ideas
and condensed matter of things

1019
00:46:49,680 --> 00:46:52,640
which have lots of randomness in them,
lots of interconnections

1020
00:46:52,640 --> 00:46:53,680
between particles.

1021
00:46:53,680 --> 00:46:57,080
This looks much more like a plain
vanilla crystal, just engineered

1022
00:46:57,080 --> 00:47:00,360
in a clever way, with some other wrinkles
that make it different from the crystals

1023
00:47:00,360 --> 00:47:03,000
we see all the time. So I'm not sure
if there's a direct connection.

1024
00:47:03,000 --> 00:47:04,920
My wife is a neuroscientist,
so if there was one,

1025
00:47:04,920 --> 00:47:07,560
I think it would create
a lot more domestic harmony.

1026
00:47:07,560 --> 00:47:10,560
But sadly, I'm not allowed to say
I'm working on her stuff just yet.

1027
00:47:12,200 --> 00:47:15,720
There is some work on trying
to look at twisted nanotubes, but,

1028
00:47:15,720 --> 00:47:17,520
you know,
one of the things that comes about

1029
00:47:17,520 --> 00:47:20,520
when you think about it is that, you know,
really this two dimensional, twisting,

1030
00:47:20,520 --> 00:47:23,040
kind of requires you
to be properly in two dimensions.

1031
00:47:23,040 --> 00:47:24,280
So when you talk about nanotubes,

1032
00:47:24,280 --> 00:47:25,040
it turns out

1033
00:47:25,040 --> 00:47:28,280
you can sort of make an argument,
but it has to be at some very, very rigid

1034
00:47:28,280 --> 00:47:31,200
specific angles.
You can sort of have this extra freedom.

1035
00:47:31,200 --> 00:47:33,360
The other thing in nanotubes is
you can get a moiré

1036
00:47:33,360 --> 00:47:35,760
by just looking at the kind of slight
layer offset.

1037
00:47:35,760 --> 00:47:37,040
Or if you take two different

1038
00:47:37,040 --> 00:47:41,560
nanotube materials like something like
that, like a copper and a tube and stuff.

1039
00:47:41,760 --> 00:47:43,560
The extent
to which that's been explored is not

1040
00:47:43,560 --> 00:47:45,480
I mean, of course
everything's been explored theoretically.

1041
00:47:45,480 --> 00:47:47,080
Right? People like to play these games.

1042
00:47:47,080 --> 00:47:49,040
People even talk about
how to do it in three dimensions,

1043
00:47:49,040 --> 00:47:52,160
because you can imagine that you do
some optical lattice where you have,

1044
00:47:52,200 --> 00:47:57,160
you know, people can engineer, motion
of neutral atoms in electron clouds.

1045
00:47:57,160 --> 00:47:59,960
When you cool atoms down,
you can make optical potentials.

1046
00:47:59,960 --> 00:48:02,960
So they move trapped in atomic worlds,
and you can do whatever

1047
00:48:03,040 --> 00:48:06,000
that's within
to some degree of whatever you like.

1048
00:48:06,000 --> 00:48:07,720
You can do whatever you like
in those samples,

1049
00:48:07,720 --> 00:48:10,320
and you can actually mimic
what happens when you twist two of them

1050
00:48:10,320 --> 00:48:11,640
or in photonic crystals.

1051
00:48:11,640 --> 00:48:12,480
So people have done that.

1052
00:48:12,480 --> 00:48:14,440
The problem
with all those kinds of platforms,

1053
00:48:14,440 --> 00:48:16,840
it's hard
to engineer strong interactions in them.

1054
00:48:16,840 --> 00:48:19,560
So while you can get that, for instance,
photonic systems actually getting strong

1055
00:48:19,560 --> 00:48:22,560
interactions in a purely photonic system
is very, very hard.

1056
00:48:22,560 --> 00:48:25,960
So you can get some more like phenomena,
but the full richness

1057
00:48:25,960 --> 00:48:27,200
seems to be hard to achieve.

1058
00:48:27,200 --> 00:48:30,200
It's possible,
but it's not obvious how you would do it.

1059
00:48:30,400 --> 00:48:31,840
Yeah, that's a great question.

1060
00:48:31,840 --> 00:48:34,400
So I'm not an experimentalist.
So you know I'm going to say

1061
00:48:34,400 --> 00:48:36,240
what I'm going to tell you
is my understanding of it.

1062
00:48:36,240 --> 00:48:38,040
So there are various ways you could do it.

1063
00:48:38,040 --> 00:48:39,400
I think the standard way is you

1064
00:48:39,400 --> 00:48:42,560
have you kind of glue these something
you kind of hold the samples on to,

1065
00:48:43,280 --> 00:48:46,200
you know, there they're various ways to
suspend these samples and stick onto them.

1066
00:48:46,200 --> 00:48:50,280
There's ways to pick them up
so they exfoliate grabs.

1067
00:48:50,280 --> 00:48:51,840
So there's this tear and stack.

1068
00:48:51,840 --> 00:48:55,640
So you tear the material, you stack it,
you lay it down, set it onto something.

1069
00:48:55,760 --> 00:48:57,800
You take the other layer
and you're using all of these

1070
00:48:57,800 --> 00:48:59,440
with two nano mechanical manipulations.

1071
00:48:59,440 --> 00:49:02,480
So you kind of know your angle
and you can twist it and drop it on there.

1072
00:49:02,720 --> 00:49:05,560
Now the success rate of this,
you know, fabulous group

1073
00:49:05,560 --> 00:49:06,920
is probably one of the best at doing it.

1074
00:49:06,920 --> 00:49:09,920
I think they have something like a 60%
success rate, something like that.

1075
00:49:10,080 --> 00:49:13,080
But typical groups starting out
start at something like 2 or 3%.

1076
00:49:13,680 --> 00:49:18,720
These are done just at microfilm with Now
the actual preparation is not usually

1077
00:49:18,720 --> 00:49:21,840
the experiments are often at millikelvin
micro Kelvin temperatures.

1078
00:49:21,840 --> 00:49:24,160
I don't know what the temperature scale
of the preparation is.

1079
00:49:24,160 --> 00:49:25,600
And now there are ways to make

1080
00:49:25,600 --> 00:49:27,240
you know, one of the things
is that originally

1081
00:49:27,240 --> 00:49:30,120
we would actually exfoliate samples,
but the best graphene is now

1082
00:49:30,120 --> 00:49:31,480
chemical vapor deposited.

1083
00:49:31,480 --> 00:49:33,440
And so it's better samples and so on.

1084
00:49:33,440 --> 00:49:35,160
So the one thing
that only one group in the world

1085
00:49:35,160 --> 00:49:36,200
does is one of the materials

1086
00:49:36,200 --> 00:49:39,240
I put there is hexagonal boron nitride,
which is just like graphene,

1087
00:49:39,360 --> 00:49:42,240
except the two carbon
atoms in graphene are replaced by boron

1088
00:49:42,240 --> 00:49:43,560
and one at one site. And,

1089
00:49:44,560 --> 00:49:45,840
nitrogen on the other.

1090
00:49:45,840 --> 00:49:48,760
That's a material that turns out
to be hard to make in the sense

1091
00:49:48,760 --> 00:49:52,200
that essentially there's one
group in the world that makes it in Japan,

1092
00:49:52,480 --> 00:49:55,160
and they supply the entire world
essentially with this thing.

1093
00:49:55,160 --> 00:49:58,440
So they're these two gentlemen,
Takashi Taniguchi and Kenji Watanabe,

1094
00:49:58,440 --> 00:50:01,080
who are on every single experimental paper
on this material.

1095
00:50:01,080 --> 00:50:04,480
If they don't give you a B and you can't
do an experiment, roughly speaking.

1096
00:50:04,760 --> 00:50:07,640
And so that's sort of that's the big
material challenge is getting good.

1097
00:50:07,640 --> 00:50:10,440
N and the graphene stuff is assembly.
Yeah.

1098
00:50:10,440 --> 00:50:11,520
So I should say one last thing.

1099
00:50:11,520 --> 00:50:14,680
There is a remarkable new experiment
done in the group of Charlie Lanni

1100
00:50:14,720 --> 00:50:16,560
who's at the Weitzman Institute in Israel.

1101
00:50:16,560 --> 00:50:19,360
What they've done is take an atomic force
microscope, which you usually think of

1102
00:50:19,360 --> 00:50:22,640
as a very sharp tip
that you use to do atomic scale forces.

1103
00:50:22,920 --> 00:50:25,200
And they did something
very counterintuitive.

1104
00:50:25,200 --> 00:50:27,120
They took that
and just chopped its head off.

1105
00:50:27,120 --> 00:50:29,160
So instead of having a sharp, pointy
pyramid,

1106
00:50:29,160 --> 00:50:32,160
they have a flat topped pyramid
with a kind of triangular profile.

1107
00:50:32,280 --> 00:50:34,440
And they can drape a graphene layer
on that pyramid.

1108
00:50:34,440 --> 00:50:36,360
And it's about, you know, what is it?

1109
00:50:36,360 --> 00:50:39,840
I think it's about 200 nanometers
by 200 nanometers in each side.

1110
00:50:40,040 --> 00:50:42,840
So you've got this flat surface
and the graphene sticks pretty well,

1111
00:50:42,840 --> 00:50:44,160
and it stays pretty flat.

1112
00:50:44,160 --> 00:50:46,080
So we've got another graphene
layer on top.

1113
00:50:46,080 --> 00:50:50,280
They can bring that layer down
and just twist so they can actually twist

1114
00:50:50,280 --> 00:50:54,080
in real time and watch all this physics
come, come in by just twisting it.

1115
00:50:54,160 --> 00:50:56,040
The experiments are just spectacular.

1116
00:50:56,040 --> 00:50:57,560
And you can use this
for all kinds of things.

1117
00:50:57,560 --> 00:50:58,800
So this is the next frontier.

1118
00:50:58,800 --> 00:51:01,040
And dozens of groups around
the world are trying to make it.

1119
00:51:01,040 --> 00:51:03,240
Now the trial has shown it can be done.

1120
00:51:03,240 --> 00:51:04,120
Oh that's fantastic.

1121
00:51:04,120 --> 00:51:06,960
So actually for graphene,
the magic angle is not completely crazy.

1122
00:51:06,960 --> 00:51:08,120
There's a calculation

1123
00:51:08,120 --> 00:51:11,920
you can do that rests on the fact
that graphene has a linear dispersion.

1124
00:51:12,360 --> 00:51:15,360
So what you can do is, you know,
maybe I'll just do that on the board.

1125
00:51:15,360 --> 00:51:17,520
So you have this picture of.

1126
00:51:17,520 --> 00:51:20,360
So this is the blue one zone of graphene.

1127
00:51:20,360 --> 00:51:23,040
And the thing you need to know about
the one zone of graphene

1128
00:51:23,040 --> 00:51:25,960
is it has this thing called a Dirac cone
at each of these corners.

1129
00:51:25,960 --> 00:51:28,760
Let me focus at one corner.
This has a linear dispersion.

1130
00:51:28,760 --> 00:51:31,960
So if I looked there
e would be like VF times.

1131
00:51:32,840 --> 00:51:33,600
Okay.

1132
00:51:33,600 --> 00:51:36,120
So now imagine I have two such layers.

1133
00:51:36,120 --> 00:51:40,280
So I have one rule one zone like this
and I have another one.

1134
00:51:40,280 --> 00:51:42,520
I always mess
this up. So let's see if I can do it.

1135
00:51:43,560 --> 00:51:45,960
That's slightly offset There we go.

1136
00:51:45,960 --> 00:51:47,640
So I've twisted them
relative to each other.

1137
00:51:47,640 --> 00:51:50,040
So I have two such cones.

1138
00:51:50,040 --> 00:51:50,440
Right.

1139
00:51:50,440 --> 00:51:53,000
You can see that I'm in the same frame.

1140
00:51:53,000 --> 00:51:53,760
So one is gone.

1141
00:51:53,760 --> 00:51:56,800
Theta over two one side one's gone
minus theta over to the other side.

1142
00:51:56,920 --> 00:51:58,280
So they started out on top of each other.

1143
00:51:58,280 --> 00:52:02,160
They moved so in momentum space,
if I drew a picture along the line

1144
00:52:02,160 --> 00:52:05,440
that joins them, I would have a picture
that looks something like this.

1145
00:52:07,120 --> 00:52:07,400
Right.

1146
00:52:07,400 --> 00:52:10,800
So this point goes here,
this point goes there.

1147
00:52:11,080 --> 00:52:14,480
And each of these has E going as VFC

1148
00:52:15,680 --> 00:52:16,600
okay.

1149
00:52:16,600 --> 00:52:20,400
So the two layers,
the only perturbation I'm going to put in

1150
00:52:20,400 --> 00:52:22,560
is when the two layers
talk to each other. We have color.

1151
00:52:23,520 --> 00:52:24,080
Excellent.

1152
00:52:24,080 --> 00:52:26,160
So let me actually make this work.

1153
00:52:26,160 --> 00:52:29,200
So let's say
I have let me just redraw these in color.

1154
00:52:29,200 --> 00:52:32,200
So the orange one came here.

1155
00:52:33,120 --> 00:52:34,120
The blue one came here.

1156
00:52:34,120 --> 00:52:36,320
Each of these are contributed
by the two layers.

1157
00:52:36,320 --> 00:52:39,600
The only potential of the moiré physics
is when the two layer each other.

1158
00:52:39,920 --> 00:52:43,520
So if I think about a perturbation theory
problem, there's not much to do here.

1159
00:52:43,520 --> 00:52:44,760
There's not much to do here.

1160
00:52:44,760 --> 00:52:46,800
But at this point
I have a degeneracy in the two layers.

1161
00:52:46,800 --> 00:52:47,840
Talk to each other.

1162
00:52:47,840 --> 00:52:49,320
So the minute I switch on a potential

1163
00:52:49,320 --> 00:52:52,320
and if you go back to your second year
quantum mechanics,

1164
00:52:52,440 --> 00:52:53,760
nature abhors a degeneracy.

1165
00:52:53,760 --> 00:52:56,800
So you'll just make a gap over
here. Right.

1166
00:52:57,480 --> 00:53:01,200
And so that is the potential let's call
that deeper.

1167
00:53:01,320 --> 00:53:03,320
That's the potential
between the layers that.

1168
00:53:03,320 --> 00:53:06,640
So this gap is of order deeper.

1169
00:53:07,400 --> 00:53:10,720
So we want to ask how far this came out.

1170
00:53:10,840 --> 00:53:12,640
Like what what was this energy.

1171
00:53:12,640 --> 00:53:15,640
This energy is let's call this distance k.

1172
00:53:15,960 --> 00:53:19,800
It turns out that that distance k is like

1173
00:53:21,160 --> 00:53:23,160
related to theta,

1174
00:53:23,160 --> 00:53:23,920
okay.

1175
00:53:23,920 --> 00:53:25,800
It's just that it's a very small scale.

1176
00:53:25,800 --> 00:53:26,600
It's related to theta.

1177
00:53:26,600 --> 00:53:29,520
It's theta times
some number that's known for graphene.

1178
00:53:29,520 --> 00:53:31,920
Let me just write it that way okay.

1179
00:53:31,920 --> 00:53:37,160
So I have to scale
I have VF times k d times theta.

1180
00:53:37,680 --> 00:53:39,960
That's how high this energy is.

1181
00:53:39,960 --> 00:53:43,440
So this is VF k d theta.

1182
00:53:43,800 --> 00:53:45,480
And I have d perp.

1183
00:53:45,480 --> 00:53:49,200
This comes from chemistry k d comes from
chemistry VF comes from chemistry.

1184
00:53:49,400 --> 00:53:50,840
Theta is in my control.

1185
00:53:50,840 --> 00:53:53,840
So if I set those two equal to each other

1186
00:53:54,600 --> 00:53:55,800
and choose theta that way.

1187
00:53:55,800 --> 00:53:58,800
So if I just say.

1188
00:53:59,280 --> 00:54:01,480
So all these numbers I know what happens

1189
00:54:01,480 --> 00:54:06,080
is that the band gets pushed down
by as much as it went up.

1190
00:54:06,560 --> 00:54:09,640
So twice to rise and gets whacked down
by exactly as much as it went up.

1191
00:54:09,920 --> 00:54:12,400
So given that I know something
from chemistry

1192
00:54:12,400 --> 00:54:14,200
about how the two layers
talk to each other,

1193
00:54:14,200 --> 00:54:17,480
and these two are numbers from
a single layer that I know from chemistry,

1194
00:54:17,880 --> 00:54:19,760
I can actually predict an angle
at where this happens.

1195
00:54:19,760 --> 00:54:21,160
And this is remarkably good.

1196
00:54:21,160 --> 00:54:23,400
That's actually an exact calculation
of the magic angle.

1197
00:54:23,400 --> 00:54:27,320
Now, the actual calculation of why
things that further terms don't push

1198
00:54:27,320 --> 00:54:30,760
things apart again and so on, requires
some details is a fancier thing.

1199
00:54:30,760 --> 00:54:31,360
But this is this

1200
00:54:31,360 --> 00:54:34,440
gets you to within the level of accuracy
I was talking about in the talk.

1201
00:54:34,440 --> 00:54:36,000
This gives you the right number. Yeah.

1202
00:54:39,280 --> 00:54:42,280
Thanks for asking.

1203
00:54:42,560 --> 00:54:44,840
This okay.

1204
00:54:44,840 --> 00:54:48,520
I think we just found out. So thank you.