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Thank you all for coming.

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So. I think the topic is,

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you all know, since presumably that played
some small role in persuading you to,

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come in.

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Like all topics in physics,

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you know, if you start to look,
there's a precursor of,

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you know, precursors of ideas
that go back a long time.

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And so topology

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already showed up in classical physics.

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You know, it seemed fluid vortices,

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the great Poincaré used topological ideas
and thinking about chaos.

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So everything does have a precursor

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till you get back to the bacteria.

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Who knows, maybe when they were using
topology in some ways that we don't

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yet understand,
but will soon, once the next model

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from open AI is out.

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But in my own

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field of condensed matter physics
and that of my colleagues today,

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we tend to think of sort of the discovery
of the quantum Hall effect,

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as a point

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where it's condensed matter physicists,

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you know, really began to have topology,

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as a main item in their arsenal.

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It existed previously in that so-called
topological sort of defects

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that we'll talk about a little bit.

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But then it really took off.

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Then it sort of went dormant for a while.

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And then there was the discovery
of these so-called topological insulators,

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which essentially extended the Hall effect

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to a much larger class
of potential phenomenon.

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All of this then goes along with
this is sort of single particle physics.

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And then there's
a lot of many body physics.

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And so on.

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So meanwhile, you know, there are parallel
developments in, in quantum

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field theory used by particle physicists.

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Again,
you know, starting in a big way in the,

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in the 70s, terms
like soliton, instantons,

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a lot of actually beautiful mathematics,
done at Oxford, with, Michael,

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and, people around him in parallel
with quantum field theory

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and of course, Roger
Penrose in the context of, other problems.

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So that's a
you should think of today as sort of a

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my job is to introduce a few ideas,

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I teach an eight week course
in the Masters in Mathematical physics.

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So this is the compressed version of it.

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This.

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Lesson.

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Right.

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And then my colleagues, who are much
more active and working on things,

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at the cutting edge at the moment
will tell you about things which are,

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more substantial in terms of,

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cutting edge physics. Very good.

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So with that, I guess all I need
is some sort of a, does this work,

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like to.

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All right.

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Okay. So

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so we start with the,

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question of, you know,
what is what is topology.

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And so of course, informally.

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Right.

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Topology studies properties of objects,

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ultimately mathematical objects

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which remain unchanged under continuous,
you know, deformations.

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So topology is prior to doing calculus,
you know, when you have

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functions that, you know, things
which have to be smooth so you can do

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things like derivatives, topology doesn't
demand that you be able to do that.

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And so of course, the example
that everybody,

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trots out is that a coffee mug

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can be continuously deformed into,
a donut.

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And here is, from Wikipedia

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for that particular maneuver.

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So as you can see, the first thing you do
is you take the coffee mug

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and you raise the,
the bottom to all the way to the top.

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It's not very useful
as a coffee mug anymore.

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And then you smooth it out and,
and there's your.

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And there's your donut. Now,

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as I said,

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you know, the mathematical
topological education of physicists,

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you know, and I belong like him
to a generation that this is not standard.

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So I can't say that I have a,

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you know, proper
understanding of the mathematics.

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I don't physicists are often described
as recreational mathematicians.

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I best, but it's not wrong.

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Pragmatic mathematicians,
maybe the other way, which is right.

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But in any event,
there is a question in mathematics.

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So when you see pictures

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like this, then these pictures
are in our three dimensional space.

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So they are sort of embedded.

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Whereas mathematicians also distinguish
between what happens

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if you talk about objects without thinking
of them as being embedded in anything.

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Right.

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So there's a sort of more abstract
ways of doing it.

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So to go one step in that direction
is that, you know,

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topology tries to classify,

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topological spaces up to homomorphisms.

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So the usual problem
of starting to read mathematics,

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which is that every definition
that under the three things

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that you realize
that you don't know, right.

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So in this case, what sort of a
logical space what's a home to a morphism.

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All right.

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So homomorphism is is a continuous
bijective map of the continuous inverse.

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We won't need it today.

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But a topological space is this.

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You know some set of points
where there's some notion of closeness.

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But you don't have the notion
of a distance. Right?

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You can't, you know,

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you don't need to have a metric
which says, I get from here to here.

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And it's, you know, this distance.

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But simply the things are close
and things are far.

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And once you do that, you can have limits,
continuity, connectedness and so on.

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So, okay, now you might say,

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why would what
what would this have to do with physics.

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So obviously in physics
units are important.

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You well you know, maybe you assume
perhaps almost all of you have been

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physics undergraduates.

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And one thing that would have been
drummed into you is units.

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How come you wrote six?

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You know what units, right?
I mean, you're a physicist.

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You have to know what units are.

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And then maybe on top of it,
dimensional analysis, right.

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And so on and so forth.

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And why do we think this
something may happen at the Planck scale?

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Well, you know, there are units.

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You combine them so and so.

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It seems a little odd to say,
let me start talking

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about stuff in physics
which doesn't have units. Right.

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So you might say,
but surely this is irrelevant.

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But what?

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So I have to sort of tell you,

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I tell you
that topology is classified spaces.

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So you might say
what spaces are interesting in physics.

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Well, so it could be real space.

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Now of course
the space we live in is what it is.

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There are cosmologists.

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I think we do wonder periodically

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whether the universe itself
is a giant donut of three dimensions.

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But we're not talking about that.

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As a condensed matter physicist,
we actually do sometimes ask the question

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of what would happen
if our condensed matter system was put

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on spaces of different topology.

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So in the case of the quantum Hall
effect, the fractional quantum

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Hall effect, you say, does it
live on a sphere?

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Does it live on a Taurus?
Does it live on a donut?

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Two holes?

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It's not something
that would happen in the laboratory,

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but it's a way of asking
a certain question about the system.

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And it turns out, for example,
that the one third quantum Hall state

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would have one state on the Taurus,
three on the sorry, one on the sphere,

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three on the Taurus,
and nine on the next one.

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And that's a way of,
discovering that that state

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which otherwise looks like a liquid,
has something special going on inside.

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Okay, the next one up is momentum space.

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So, this requires that you have been
forced to sit through solid state physics.

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I hope you were,

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and so you may

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remember things called Fermi surfaces
and filament zones.

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So the one,

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the spherical one is that of sodium.

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And so that just looks like a sphere.

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And notice that it sits inside
this wire cage, which is the balloon zone.

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So that's the unit in which
things are going to repeat.

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And momentum space. So you have one sphere
in a second and a third.

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But they won't touch each other.

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But the next one for copper
has this black things

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which are actually pipes
connecting to the next bell, ones

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which have been cut off to display it.

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So if we actually plotted the whole thing,
you would have this right thing

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extending through this momentum space

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with all these tubes
connecting neighboring cells.

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So you can see that
the topology of the two is different.

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In one case that,

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you know, you can surround the sphere
and disconnected from everybody else.

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In the other case,
they're just pipes all over the place.

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Looks like the basement of a particularly,
you know, difficult building.

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Okay.

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Then you could say, well, what about,
you know, spaces of trajectories.

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So this will come up in, in, Steve's talk
that you have, you should think of these

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as identical particles, quantum
mechanically identical particles.

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So here they are at initially
and here they are some later time.

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And along the way,

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if we insist that they don't,
they cannot be at the same point

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at the same time,
which were fermions of standard.

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But let me say
there are other particles, and indeed

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these will be the anyons of this talk
for which this is true.

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Then you can sort of see that this is like
taking bits of string and, you know,

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kind of maybe braiding them around another
and reconnecting them at the top.

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And, you know, you can't simply
there's something stable about that,

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because once you've scrambled them up
a bit, braided them a bit,

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you cannot unbreak them without really
taking them off the hooks at the top.

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And so the space of trajectories
can be another one which has topologies.

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And that will show up today

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and then up the food chain,
the things called fiber bundles.

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And while this picture is useless
for the purposes of actually

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doing anything with fiber bundles really,
but it is roughly the idea

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which is you have some base manifold

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and then these things
which have been drawn as fibers, but

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the main point is
there is a space for physics.

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You know,
it could be, for example, real space.

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And the fiber is
something that lives at that point.

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So you've all done Maxwell theory
electric and magnetic fields.

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So think of real space
as the base manifold and the electric

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field at the point, or the magnetic field
at a point as living in the fiber.

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And then that's the that's the space.

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And it could be space.
It could be space time.

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So so this could be a space
whose topology we're interested in.

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Then you

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could have extended objects

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in condensed matter physics
that a vertex lines, vertex loops,

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you could even make them up
in gauge theories

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that are things like a Wilson loop
operators, extended operators.

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And in fact, again

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and again, Steve may not bring them up,
but some of the things we'll talk about,

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you calculate them
mathematically as Wilson loop operators.

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And you're interested in questions
such as do they link or do they not?

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And actually do they not.

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Then you know Ode,
which is a beautiful circuit.

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All right.

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So that's

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basically to to explain
that there are lots of settings in which

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you would be interested in the topology
of some space that is relevant to physics.

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That doesn't mean that

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the physics of distances
has gone away of units that's there.

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So sometimes there is a specific term
that you're trying to calculate

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which is not distance dependent,
but is stuff topology dependent.

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And occasionally you get lucky.

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And the whole answer depends
upon the topology and nothing else.

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But that's, you know, more rare
and of course,

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more beautiful when that happens.

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All right. So

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as a subject in

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mathematics, topology has all sorts of,
you know, tricks up its sleeve.

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So, of course, ideally
you could just be completely confident

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whether there's a morphism or not,
but topologies, especially that talk about

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very, very abstractly
defined spaces have come up with ways

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of thinking about them,
which are somewhere partial.

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You can decide whether two spaces are

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the same or not, depending on
asking some more limited question.

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So one of the techniques, goes

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under the topic of homotopy groups.

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This is actually not the one

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that turns out to be
mathematically the easiest to do,

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but it's the one that's easiest
to explain.

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So basically in homotopy, what you do

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is you have your space sitting somewhere
that you're interested in,

244
00:11:59,720 --> 00:12:03,840
and on the right hand side you put down
spheres of various dimensions.

245
00:12:04,080 --> 00:12:04,440
Right.

246
00:12:04,440 --> 00:12:07,560
So let's start with the simplest,
but it's not the simplest one.

247
00:12:07,560 --> 00:12:09,400
But let's start
with the second simplest one,

248
00:12:09,400 --> 00:12:12,240
which is a circle,
then the sphere in our dimensions.

249
00:12:12,240 --> 00:12:15,000
And then of course you go up and one more
dimension and one more dimension from

250
00:12:16,320 --> 00:12:18,720
having done that, you consider a map

251
00:12:18,720 --> 00:12:22,800
from this thing that you put down here
to the space of interest.

252
00:12:22,840 --> 00:12:23,040
Right.

253
00:12:23,040 --> 00:12:24,840
So there's some functions you can write,

254
00:12:24,840 --> 00:12:27,720
and you want these functions
to be continuous.

255
00:12:27,720 --> 00:12:31,160
But then you ask the question,
how many different types of functions are

256
00:12:31,920 --> 00:12:34,920
and what's the definition of type
or equivalence class.

257
00:12:35,120 --> 00:12:37,680
It's that if you can deform one function

258
00:12:37,680 --> 00:12:41,800
continuously into the other,
you say they're the same, right?

259
00:12:41,800 --> 00:12:45,680
And if you can't,
now you have a new art. So a

260
00:12:46,680 --> 00:12:48,760
this is a, you know, sacred maneuver

261
00:12:48,760 --> 00:12:52,480
in mathematics that you look
at equivalence classes of objects.

262
00:12:52,840 --> 00:12:56,280
And having done that, then you're often
interested in how do the equivalence

263
00:12:56,280 --> 00:12:59,800
classes do they compose,
can you multiply them?

264
00:12:59,800 --> 00:13:01,280
You know, and so on and so forth.

265
00:13:01,280 --> 00:13:03,000
And in the case of homotopy, which,

266
00:13:04,120 --> 00:13:05,880
it will turn out that

267
00:13:05,880 --> 00:13:09,440
these equivalence classes
will have a group structure.

268
00:13:09,800 --> 00:13:14,280
And if you go from the sphere and n
dimensions to, space,

269
00:13:14,280 --> 00:13:17,280
those are called the that will be called
the Anscombe or Tobiko.

270
00:13:17,760 --> 00:13:18,080
All right.

271
00:13:18,080 --> 00:13:21,840
So we will do use the first
and the second, which are

272
00:13:22,160 --> 00:13:25,080
and then and then stop.

273
00:13:25,080 --> 00:13:28,120
I know a little bit about PC, but
after that I have no clue what's going on.

274
00:13:29,080 --> 00:13:29,880
All right.

275
00:13:29,880 --> 00:13:33,520
So if you're mapping from a circle

276
00:13:33,800 --> 00:13:36,800
to some space, so let's say the space is,
you know, sort of I on

277
00:13:37,440 --> 00:13:40,880
a map from the circle is to draw
a loop on the space that you are going to.

278
00:13:41,640 --> 00:13:44,640
That's, that's essentially what it means,
because each point is mapping somewhere

279
00:13:44,840 --> 00:13:47,840
because the circle closes on itself.

280
00:13:47,840 --> 00:13:50,840
Any loop that I draw in
the space has to close on itself.

281
00:13:51,080 --> 00:13:52,120
So far, so good.

282
00:13:52,120 --> 00:13:56,440
So what we're asking is you have this
space, unknown space, you call it X.

283
00:13:56,880 --> 00:13:59,200
How many different types
of loops can you draw on it?

284
00:14:00,280 --> 00:14:01,280
That is

285
00:14:01,280 --> 00:14:04,280
what the first homotopy group
is going to tell you.

286
00:14:04,600 --> 00:14:09,200
Okay,
so if we are looking on the plane right.

287
00:14:09,200 --> 00:14:12,200
So if let's say on the blackboard,
if I draw something,

288
00:14:12,760 --> 00:14:15,480
you can always shrink it to zero, right.

289
00:14:15,480 --> 00:14:17,720
It doesn't matter what complicated thing
it is. Think.

290
00:14:17,720 --> 00:14:20,960
You know, it just sort of continuously
keeps shrinking is shrinking shrinking.

291
00:14:21,160 --> 00:14:23,880
It will be down to a point.
Then you can get rid of it.

292
00:14:23,880 --> 00:14:25,160
But something interesting happens

293
00:14:25,160 --> 00:14:28,160
if you decide to exclude
one point from the plane.

294
00:14:28,760 --> 00:14:29,920
So here it is.

295
00:14:29,920 --> 00:14:31,080
The black dot is gone.

296
00:14:31,080 --> 00:14:32,480
It's a hole in the plane.

297
00:14:32,480 --> 00:14:37,840
So now I could have had a curve here
that I could shrink to zero.

298
00:14:37,840 --> 00:14:41,160
But the moment I've gone once around it,
it cannot be shrunk to zero.

299
00:14:41,280 --> 00:14:44,560
If I go twice, I can't shrink it to this,
I can't deform it to this.

300
00:14:44,840 --> 00:14:45,920
And likewise right.

301
00:14:45,920 --> 00:14:49,720
So the number of times
you think of carrying a piece of rope

302
00:14:49,960 --> 00:14:51,240
and walking around

303
00:14:51,240 --> 00:14:53,960
and sort of stretching it out, right,
and then tying it at the end,

304
00:14:53,960 --> 00:14:57,480
as long as you are, you are living on
the plane yourself, right?

305
00:14:57,880 --> 00:14:59,680
You're not allowed to jump over it.

306
00:14:59,680 --> 00:15:02,080
You know, you can't unwind it, right
once you've tied it.

307
00:15:02,080 --> 00:15:04,760
So that tells you
that the first homotopy group

308
00:15:04,760 --> 00:15:08,120
of the plane with the point
removed is the integers.

309
00:15:08,120 --> 00:15:09,360
Because. So.

310
00:15:09,360 --> 00:15:12,760
And then to compose them you just do one
and then you do the second following.

311
00:15:12,760 --> 00:15:13,960
And that's called multiplication.

312
00:15:13,960 --> 00:15:16,960
And in that it looks like addition here.

313
00:15:17,280 --> 00:15:18,520
Sorry I seem to have lost.

314
00:15:18,520 --> 00:15:19,320
Oh sorry.

315
00:15:19,320 --> 00:15:21,600
The last line it says the following.

316
00:15:21,600 --> 00:15:23,680
I said it for a plane
with the point removed,

317
00:15:23,680 --> 00:15:25,800
but actually
most of the plane was irrelevant.

318
00:15:25,800 --> 00:15:27,960
I can just shrink the rest of the plane.

319
00:15:27,960 --> 00:15:32,120
Basically just talk about a circle,
I can expand the whole and so on.

320
00:15:32,120 --> 00:15:34,760
And so if you say you're going to

321
00:15:35,920 --> 00:15:37,080
draw loops

322
00:15:37,080 --> 00:15:41,040
on a circle, that's also
the same classification, right?

323
00:15:41,120 --> 00:15:42,120
Go around the circle.

324
00:15:42,120 --> 00:15:42,400
You know,

325
00:15:42,400 --> 00:15:44,800
you don't go around, go around ones
go around twice or go around

326
00:15:44,800 --> 00:15:47,040
minus one on the other side
minus two the other side.

327
00:15:47,040 --> 00:15:48,040
So far so good.

328
00:15:48,040 --> 00:15:50,760
So that's the meaning of saying that maps
from the circle to

329
00:15:50,760 --> 00:15:53,760
the circle are classified by the integers.

330
00:15:53,960 --> 00:15:58,200
So this is a simple example
of a topological construct.

331
00:15:58,200 --> 00:16:00,520
What can we use this one.

332
00:16:00,520 --> 00:16:01,600
So we can ask matter.

333
00:16:01,600 --> 00:16:06,600
Physicists are often interested
in things that look like magnets, either

334
00:16:06,600 --> 00:16:09,680
because they are magnets
or because we've learned to map

335
00:16:09,680 --> 00:16:12,680
all sorts of things on to things
that look like magnets.

336
00:16:13,320 --> 00:16:15,600
And so often we talk about models.

337
00:16:15,600 --> 00:16:18,600
So for example,
there's something called the x, y model.

338
00:16:18,600 --> 00:16:22,000
And what the x y model is is x y
because there's a spin.

339
00:16:22,280 --> 00:16:23,720
And the spin has to live in a plane,

340
00:16:23,720 --> 00:16:26,720
which we call the xy plane,
therefore the x y model.

341
00:16:27,000 --> 00:16:30,840
And so now a ground state,

342
00:16:33,160 --> 00:16:34,480
in a magnet

343
00:16:34,480 --> 00:16:38,520
ferromagnetic is one in which
all the spins agree on what to do, right?

344
00:16:38,720 --> 00:16:40,720
So every spin will point
in the same direction.

345
00:16:40,720 --> 00:16:43,480
So you, the A picture in just a second.

346
00:16:43,480 --> 00:16:46,480
But then if the energy is not minimum,

347
00:16:46,680 --> 00:16:51,600
the spins will deviate from this
exceptionally happy configuration.

348
00:16:51,960 --> 00:16:54,120
Right? So many things can happen.

349
00:16:54,120 --> 00:16:55,360
But one of the things that can happen are

350
00:16:55,360 --> 00:16:58,360
what are called topological defects,
which I'll just describe in a minute.

351
00:16:58,840 --> 00:17:02,880
And the claim is in the XY model,
the topological defects are classified

352
00:17:02,880 --> 00:17:04,280
by pi one of s one.

353
00:17:04,280 --> 00:17:07,720
The first homotopy group of the circle.

354
00:17:07,720 --> 00:17:09,520
So how does this happen.

355
00:17:09,520 --> 00:17:11,360
So here's the ground state right.

356
00:17:11,360 --> 00:17:13,560
So everybody's pointing the same way.

357
00:17:13,560 --> 00:17:17,080
Now this is an example
of a spontaneously broken symmetry.

358
00:17:17,920 --> 00:17:20,720
Which means that while I've drawn it
this way

359
00:17:20,720 --> 00:17:23,480
really we could globally rotate
every single spin.

360
00:17:23,480 --> 00:17:25,600
You can just take the picture,
do this, do that, do that.

361
00:17:25,600 --> 00:17:28,400
And that's an equally allowed
configuration, right?

362
00:17:28,400 --> 00:17:32,080
I'm imagining rotating the spins,
not the space in which they live.

363
00:17:32,560 --> 00:17:35,560
So that axis,
the fact that they're pointing up

364
00:17:35,920 --> 00:17:38,840
is just a particular choice really.

365
00:17:38,840 --> 00:17:40,080
They could point in any direction.

366
00:17:40,080 --> 00:17:43,080
But the important thing is
that in the ground state, they all agree.

367
00:17:43,080 --> 00:17:47,040
The energetics of a magnet of a ferro
magnet says to each spin,

368
00:17:47,280 --> 00:17:48,680
try and agree with your neighbors.

369
00:17:50,760 --> 00:17:51,600
Okay.

370
00:17:51,600 --> 00:17:53,200
And then antiferromagnetic.

371
00:17:53,200 --> 00:17:54,680
In the simplest case,

372
00:17:54,680 --> 00:17:57,680
the instruction to each spin
is try and disagree with your neighbors.

373
00:17:58,440 --> 00:18:01,440
Which, looks a bit
more like American politics.

374
00:18:03,720 --> 00:18:05,600
Which which is my country of citizenship.

375
00:18:05,600 --> 00:18:07,520
So I'm allowed to.

376
00:18:07,520 --> 00:18:09,760
Okay, good.

377
00:18:09,760 --> 00:18:11,880
Very good. Now,

378
00:18:11,880 --> 00:18:15,480
you could make deformations of this
if I said, you know, raise the energy.

379
00:18:15,480 --> 00:18:16,800
You could say it could take a given spin

380
00:18:16,800 --> 00:18:18,360
and you could move it a little bit,
and you would say,

381
00:18:18,360 --> 00:18:20,840
I should expect
to end up at a higher energy anyway.

382
00:18:20,840 --> 00:18:24,080
But there are particularly
interesting things which you can do

383
00:18:24,320 --> 00:18:25,960
which are very hard to get rid of.

384
00:18:25,960 --> 00:18:28,440
And that's where the topology comes
in. Okay.

385
00:18:28,440 --> 00:18:31,480
So here's a vertex right.

386
00:18:31,480 --> 00:18:33,000
So of course you know them from fluids.

387
00:18:33,000 --> 00:18:36,000
But I want to explain what this has to do
with the topology that I was describing.

388
00:18:36,840 --> 00:18:41,280
So the first thing is a, a a defect

389
00:18:42,600 --> 00:18:45,720
by its name suggests something doesn't
look right somewhere.

390
00:18:45,720 --> 00:18:47,720
So in this case something doesn't look
right at the middle.

391
00:18:47,720 --> 00:18:49,440
So let's ignore that.

392
00:18:49,440 --> 00:18:53,800
If we go far from the nominal location
of the defect, what we expect

393
00:18:53,800 --> 00:18:57,800
to find on energetic grounds
is that, at least locally,

394
00:18:59,360 --> 00:19:01,000
this configuration should look like

395
00:19:01,000 --> 00:19:04,000
it's in a ground state, because that's
the state of minimum energy.

396
00:19:04,640 --> 00:19:07,640
And indeed, if you look at this picture,
you will notice that if you

397
00:19:07,640 --> 00:19:09,560
I only gave you a small snapshot of this,

398
00:19:09,560 --> 00:19:11,680
you would say
the spins are all pointing up.

399
00:19:11,680 --> 00:19:15,840
If I said this, you would say to the left
down to the right, right.

400
00:19:16,120 --> 00:19:19,000
So locally I have something
that looks like

401
00:19:19,000 --> 00:19:22,440
a ground state,
but there's this aspect that globally,

402
00:19:23,440 --> 00:19:26,560
even very
far the spins are not all agreeing.

403
00:19:26,600 --> 00:19:29,600
Right? So they sort of slowly twist.

404
00:19:30,120 --> 00:19:32,400
And in fact,
we could do it mathematically.

405
00:19:32,400 --> 00:19:33,720
If we go further and further out,

406
00:19:33,720 --> 00:19:36,720
we can write down the configuration
in which that, you know,

407
00:19:36,800 --> 00:19:39,040
energy can be calculated.

408
00:19:39,040 --> 00:19:42,360
So given the assumption

409
00:19:43,200 --> 00:19:45,360
which is energetically justified,

410
00:19:45,360 --> 00:19:48,360
that far from the location of the defect,

411
00:19:48,480 --> 00:19:51,480
the system must locally be
in the ground state.

412
00:19:51,520 --> 00:19:52,960
We can now ask the following question.

413
00:19:52,960 --> 00:19:57,000
We can say go very far away
and walk around the basics.

414
00:19:57,520 --> 00:19:59,160
What do you see?

415
00:19:59,160 --> 00:20:01,920
So observe
that you're walking in a circle.

416
00:20:01,920 --> 00:20:06,280
That's your S1
from which you are mapping locally.

417
00:20:06,280 --> 00:20:07,720
You see a direction of the spin.

418
00:20:07,720 --> 00:20:09,720
The spin itself lives on a circle.

419
00:20:09,720 --> 00:20:12,640
That's the S1 to which you are mapping.

420
00:20:12,640 --> 00:20:13,520
And what do you see?

421
00:20:13,520 --> 00:20:16,760
If you walk clockwise,
you find that the spin rotates.

422
00:20:16,760 --> 00:20:18,000
Sorry, anti-clockwise.

423
00:20:18,000 --> 00:20:20,560
The spin rotates anti-clockwise,
of course.

424
00:20:20,560 --> 00:20:22,320
Clockwise, clockwise. Right.

425
00:20:23,280 --> 00:20:24,920
If you go back to the ground state

426
00:20:24,920 --> 00:20:27,920
and we do the same thing at infinity,
what do we see?

427
00:20:28,400 --> 00:20:31,000
The spin doesn't rotate at all.

428
00:20:31,000 --> 00:20:33,480
So the ground state is characterized

429
00:20:33,480 --> 00:20:37,040
by the trivial map in which every point.

430
00:20:37,040 --> 00:20:39,840
Maybe I should just draw this here.

431
00:20:39,840 --> 00:20:43,000
So this is the S1 from which I'm mapping.

432
00:20:43,000 --> 00:20:44,760
This is the S1 which I'm mapping.

433
00:20:44,760 --> 00:20:48,000
So the ground state consists
of mapping every point

434
00:20:48,000 --> 00:20:51,000
on this circle
to a single point in that circle.

435
00:20:51,440 --> 00:20:54,440
But the vertex consists of mapping

436
00:20:54,680 --> 00:20:57,240
every point right.

437
00:20:57,240 --> 00:21:00,360
And you'll see the same recur
to the identical point.

438
00:21:01,640 --> 00:21:02,280
Right.

439
00:21:02,280 --> 00:21:04,920
And these two are in different
topological classes.

440
00:21:04,920 --> 00:21:08,360
Small deformations of these maps
cannot take you from one to the other.

441
00:21:08,360 --> 00:21:10,000
You have to do something pretty major.

442
00:21:10,000 --> 00:21:12,720
You have to take your string
around the whole are you don't right?

443
00:21:12,720 --> 00:21:14,720
And you have to cut it in order to do it.

444
00:21:14,720 --> 00:21:16,240
That so.

445
00:21:17,640 --> 00:21:20,040
So what you can say is quite generally

446
00:21:20,040 --> 00:21:23,160
that if you have an XYZ system
and you are in two dimensions

447
00:21:23,160 --> 00:21:27,080
and you have a defect,
if you walk around it at very far away,

448
00:21:27,360 --> 00:21:30,960
you should be picking up
some element of the homotopy group.

449
00:21:31,360 --> 00:21:33,960
We've seen, you know, zero and one.

450
00:21:33,960 --> 00:21:39,160
And if you want to know what two
looks like, well here it is again.

451
00:21:39,200 --> 00:21:40,760
Don't worry
about what happens in the middle.

452
00:21:40,760 --> 00:21:42,840
Just focus on
what happens at the boundary.

453
00:21:42,840 --> 00:21:45,080
It's points to the right, left.

454
00:21:45,080 --> 00:21:46,800
See that is returned to itself.

455
00:21:46,800 --> 00:21:48,480
Although only gone halfway.

456
00:21:48,480 --> 00:21:49,480
And then it will repeat.

457
00:21:49,480 --> 00:21:53,080
So it will rotate twice
for each circle at infinity.

458
00:21:53,080 --> 00:21:57,320
And that will become the element
corresponding to the second.

459
00:21:57,360 --> 00:21:59,120
You know, the instead of being two.

460
00:21:59,120 --> 00:22:02,760
And then I can draw pictures in which,
you know, you go clockwise and the spin

461
00:22:02,760 --> 00:22:06,800
rotates anti-clockwise and then that will
give you negative numbers. So

462
00:22:07,800 --> 00:22:09,000
now the reason that

463
00:22:09,000 --> 00:22:12,200
the topological stuff is interesting
is that because we know

464
00:22:12,200 --> 00:22:15,320
that you can't continuously deform
one of these maps into the other,

465
00:22:16,080 --> 00:22:17,040
it shows you that

466
00:22:17,040 --> 00:22:20,160
if you had such a configuration
and you looked at it at infinity,

467
00:22:20,520 --> 00:22:23,520
you could be confident
that you can't just, you know,

468
00:22:23,960 --> 00:22:26,960
by flipping a few spins,
you can't get to the other one.

469
00:22:27,080 --> 00:22:30,840
You've got to do something very,
you know, large scale now

470
00:22:31,440 --> 00:22:33,080
in physics, you know, the energy of

471
00:22:33,080 --> 00:22:35,720
if you have a system of a certain size,
the energy is something finite.

472
00:22:35,720 --> 00:22:37,440
So sometimes I like to say that, you know,

473
00:22:37,440 --> 00:22:41,200
topology proposes but energetics disposes
to just being physics.

474
00:22:41,200 --> 00:22:41,920
Right.

475
00:22:41,920 --> 00:22:42,720
And that's true.

476
00:22:42,720 --> 00:22:44,800
But nevertheless topology
gives you a very good idea

477
00:22:44,800 --> 00:22:47,320
that some things may be very,
very hard to do.

478
00:22:47,320 --> 00:22:47,880
Okay.

479
00:22:47,880 --> 00:22:51,560
So that's now there's a much fancier
topological theory of defects, which,

480
00:22:51,600 --> 00:22:54,000
you know,
liquid crystals, things like that.

481
00:22:54,000 --> 00:22:56,400
Obviously the utility comes
when you use mathematics

482
00:22:56,400 --> 00:22:57,720
to find things that, you know,
you couldn't

483
00:22:57,720 --> 00:23:00,200
just have said
by just looking at pictures,

484
00:23:00,200 --> 00:23:03,320
and, and trust me that, it's examples
like that.

485
00:23:04,160 --> 00:23:06,480
All right. Okay.

486
00:23:06,480 --> 00:23:09,760
So I want to use this,
homotopy for one more thing.

487
00:23:10,720 --> 00:23:12,520
And are you all able to see

488
00:23:12,520 --> 00:23:15,520
the board more than. No.

489
00:23:16,560 --> 00:23:19,560
Because I think you need eyes.

490
00:23:19,920 --> 00:23:21,120
Okay.

491
00:23:21,120 --> 00:23:24,120
And where other the,

492
00:23:25,000 --> 00:23:26,480
this is good.

493
00:23:26,480 --> 00:23:28,080
All right.

494
00:23:28,080 --> 00:23:30,480
So let's let's do that.

495
00:23:30,480 --> 00:23:31,080
Okay.

496
00:23:31,080 --> 00:23:33,360
So so far
this was kind of classical, right?

497
00:23:33,360 --> 00:23:35,720
Classical in the sense
I was just doing energetics.

498
00:23:35,720 --> 00:23:39,960
I now want to show you that the same sort
of stuff can enter quantum mechanics.

499
00:23:39,960 --> 00:23:42,960
So I'm going to consider
a particle on a circle.

500
00:23:44,960 --> 00:23:45,840
Is this readable.

501
00:23:45,840 --> 00:23:48,840
No not really. Okay, okay. What?

502
00:23:48,960 --> 00:23:52,800
Okay, so
this is why I mostly opted to do slides.

503
00:23:53,360 --> 00:23:54,600
And you don't really need this.

504
00:23:54,600 --> 00:23:55,800
It's a particle on a circle.

505
00:23:55,800 --> 00:23:58,280
Let me try and

506
00:23:58,280 --> 00:24:00,920
do something bigger.

507
00:24:00,920 --> 00:24:02,200
All right.

508
00:24:02,200 --> 00:24:04,320
How about that? Any better?

509
00:24:04,320 --> 00:24:06,120
Yes. Okay. Good. Excellent.

510
00:24:07,840 --> 00:24:09,360
When you all this enthusiastic

511
00:24:09,360 --> 00:24:12,360
when you were in classes as young people.

512
00:24:14,360 --> 00:24:16,440
All right, so

513
00:24:16,440 --> 00:24:18,080
here is the location of the particle.

514
00:24:18,080 --> 00:24:21,400
Just to simplify,
let's take the circle to have radius one

515
00:24:22,720 --> 00:24:23,040
okay.

516
00:24:23,040 --> 00:24:26,040
So the circumference has lengths two pi.

517
00:24:26,680 --> 00:24:27,720
The particle is here.

518
00:24:27,720 --> 00:24:32,000
And we will say well at time
t it's at some location theta of t.

519
00:24:33,000 --> 00:24:37,320
And then it's going to go at from
theta of zero is going to be some,

520
00:24:37,440 --> 00:24:42,800
some initial theta and theta of
T is going to be some final theta.

521
00:24:42,800 --> 00:24:46,200
So we're going to start somewhere
and somewhere right okay.

522
00:24:46,920 --> 00:24:50,280
So in classical mechanics

523
00:24:51,360 --> 00:24:53,640
we're just going to do a free particle.

524
00:24:53,640 --> 00:24:57,320
And since the radius is fixed
and one you know distance and angle

525
00:24:57,320 --> 00:24:59,880
are the same thing.
Although the units have to be different.

526
00:25:01,320 --> 00:25:04,640
So when you write your homework solutions
I will penalize you for those.

527
00:25:04,640 --> 00:25:08,080
But I myself
am going to take ignore units.

528
00:25:09,240 --> 00:25:10,760
All right. Good.

529
00:25:10,760 --> 00:25:14,520
So that's our free particle Lagrangian
half mass times the velocity squared.

530
00:25:15,040 --> 00:25:17,640
And as you'll recall

531
00:25:17,640 --> 00:25:20,640
what we want to do
is that once we write the action

532
00:25:20,880 --> 00:25:25,560
and then extrema
is it subject to these two conditions.

533
00:25:25,680 --> 00:25:26,280
Right.

534
00:25:26,280 --> 00:25:29,280
So that's that's classical mechanics.

535
00:25:29,360 --> 00:25:32,480
Now we can derive
the equations of motion from this.

536
00:25:32,480 --> 00:25:35,800
And they're just going to say
that the mass times

537
00:25:35,800 --> 00:25:38,800
acceleration has to be zero
because it's a free particle.

538
00:25:40,280 --> 00:25:42,400
All right.

539
00:25:42,400 --> 00:25:45,560
Now one thing about the particle

540
00:25:45,560 --> 00:25:48,960
living in the circle
that we just discussed is that

541
00:25:50,640 --> 00:25:53,160
it can start,

542
00:25:53,160 --> 00:25:55,080
you know, so let's say this is theta

543
00:25:55,080 --> 00:25:58,080
I and this is theta f.

544
00:25:58,480 --> 00:25:59,760
So you can certainly do this.

545
00:26:00,800 --> 00:26:02,720
But it could also

546
00:26:02,720 --> 00:26:05,960
do that or go around
a certain number of times.

547
00:26:05,960 --> 00:26:06,200
Right.

548
00:26:06,200 --> 00:26:09,680
So actually this is one of these cases
where you know,

549
00:26:10,520 --> 00:26:13,520
many trajectories are possible for the,

550
00:26:16,000 --> 00:26:19,360
same sort of initial and final conditions.

551
00:26:19,840 --> 00:26:22,760
In fact, sometimes it's simple
to simply take the final condition

552
00:26:22,760 --> 00:26:23,800
to be the initial condition.

553
00:26:23,800 --> 00:26:28,160
Just for analysis, which gives there's
a solution in which you just sit there.

554
00:26:28,240 --> 00:26:30,280
That's the bureaucratic solution.

555
00:26:30,280 --> 00:26:33,280
And then there are solutions of the,

556
00:26:33,840 --> 00:26:36,840
I just I get the feeling that,

557
00:26:37,280 --> 00:26:40,720
this particular audience is not as hostile
to bureaucratic,

558
00:26:41,840 --> 00:26:44,840
inertia as, some of us have come to be.

559
00:26:45,160 --> 00:26:46,120
All right.

560
00:26:46,120 --> 00:26:49,120
I can define
something called the winding number,

561
00:26:49,160 --> 00:26:52,320
which is I just integrate the time

562
00:26:52,320 --> 00:26:55,320
derivative right of theta.

563
00:26:55,560 --> 00:26:57,440
Now, you know,
you might be tempted to say,

564
00:26:57,440 --> 00:27:01,440
well, this is just theta f minus theta I,
but because this is a periodic variable,

565
00:27:01,440 --> 00:27:02,520
you know, this is not true.

566
00:27:02,520 --> 00:27:06,640
It's actually this plus two pi times L,
which is the winding numbers.

567
00:27:07,120 --> 00:27:07,520
Right.

568
00:27:07,520 --> 00:27:11,000
So that's because it's a multiple
multiple valued variable.

569
00:27:11,880 --> 00:27:13,840
Good. So now I claim

570
00:27:15,080 --> 00:27:17,400
that I can change this action

571
00:27:17,400 --> 00:27:20,400
to this plus some number l five times w.

572
00:27:21,680 --> 00:27:24,400
And that that's absolutely every bit
as good

573
00:27:24,400 --> 00:27:27,400
a classical action
as the one that I wrote down at the start.

574
00:27:28,080 --> 00:27:31,400
Okay, so if you remember dimly,

575
00:27:32,880 --> 00:27:35,040
you know, you may remember that

576
00:27:35,040 --> 00:27:37,800
total derivatives don't matter for action.

577
00:27:37,800 --> 00:27:38,880
If not,

578
00:27:38,880 --> 00:27:41,560
the next thing you can say is, well,
if I have something like this, which I'm

579
00:27:41,560 --> 00:27:43,120
keeping the end point fixed,

580
00:27:43,120 --> 00:27:46,240
something that seems to depend only
on the end points, you know, can't matter

581
00:27:46,240 --> 00:27:49,240
because I can do my variation
without this value changing.

582
00:27:49,360 --> 00:27:52,800
But in the worst case, you just pull out
your Euler-Lagrange equations

583
00:27:53,280 --> 00:27:56,000
and you differentiate
with respect to theta,

584
00:27:56,000 --> 00:27:58,920
not you get a constant differentiate
with respect to time you get zero.

585
00:27:58,920 --> 00:28:00,840
So literally you get the same equation.

586
00:28:00,840 --> 00:28:02,880
So classical mechanics doesn't care

587
00:28:02,880 --> 00:28:06,480
whether I do this action or I add to it
something that looks like that.

588
00:28:07,440 --> 00:28:09,880
But quantum mechanics

589
00:28:09,880 --> 00:28:12,000
isn't indifferent,
because in quantum mechanics,

590
00:28:12,000 --> 00:28:15,040
what I'm going to do
is I'm going to write a path integral

591
00:28:15,480 --> 00:28:18,480
which will take me from theta I to theta F

592
00:28:18,600 --> 00:28:22,160
right in summertime t,
and then I have to let me

593
00:28:22,160 --> 00:28:25,160
put in this number alpha
because it will matter.

594
00:28:25,200 --> 00:28:28,720
And the reason
is that what you're going to do is,

595
00:28:31,680 --> 00:28:34,400
you're going to say, well,

596
00:28:34,400 --> 00:28:37,400
this is a sum over all parts

597
00:28:38,200 --> 00:28:41,880
of E to the I s plus alpha times w.

598
00:28:42,680 --> 00:28:45,680
Now what I'll do is I'll take my parts
and I'll break them up

599
00:28:45,680 --> 00:28:49,200
into parts of a given homotopy right,
a given winding number.

600
00:28:49,680 --> 00:28:54,120
And so I can say,
well this thing is sum over all winding

601
00:28:54,120 --> 00:28:58,800
numbers e to the I alpha times
the winding number, sum over the parts

602
00:28:59,640 --> 00:29:02,760
that belong
to a winding number of e to the I s.

603
00:29:03,560 --> 00:29:06,720
And as you can see, this set of numbers

604
00:29:06,720 --> 00:29:08,880
right is going to be the same
no matter what alpha is.

605
00:29:08,880 --> 00:29:11,520
But when I put an alpha,
it obviously makes a different.

606
00:29:11,520 --> 00:29:13,760
So what I discovered
is that when I quantize

607
00:29:14,760 --> 00:29:17,520
this fact that my space was

608
00:29:17,520 --> 00:29:22,280
not fully connected, suddenly
pops up and tells me, you know what?

609
00:29:22,280 --> 00:29:24,000
Your quantum mechanics is ambiguous.

610
00:29:24,000 --> 00:29:26,160
You have choices to make.

611
00:29:26,160 --> 00:29:29,280
And those choices are sensitive
to this winding number.

612
00:29:29,880 --> 00:29:31,840
Now, of course,
you know this in some other language.

613
00:29:31,840 --> 00:29:34,360
If I put a magnetic flux through the ring,

614
00:29:34,360 --> 00:29:37,840
right, it's not in the ring, it's
not on the ring, but it's in the ring.

615
00:29:37,840 --> 00:29:39,960
And that's the wrong effect,
which says that, you know,

616
00:29:39,960 --> 00:29:42,920
that flux is something that the
that the particle sees.

617
00:29:42,920 --> 00:29:45,360
But this is something
you can discover from topology.

618
00:29:45,360 --> 00:29:48,600
And then there are fancier field theoretic
examples with stuff, illogical terms

619
00:29:48,960 --> 00:29:51,400
which do not look like
putting a flux through anything.

620
00:29:51,400 --> 00:29:54,000
But you simply observe
that the space is topology.

621
00:29:54,000 --> 00:29:55,600
You can write down such a term,

622
00:29:55,600 --> 00:29:58,560
and now you know that quantum mechanics
can have such things, right?

623
00:29:58,560 --> 00:30:01,560
The most famous example of
that is the theta term in QCD,

624
00:30:01,560 --> 00:30:05,240
which is supposed to have all sorts
of fascinating possibilities,

625
00:30:05,240 --> 00:30:08,400
although the state of the discussion
involves axioms

626
00:30:08,400 --> 00:30:13,440
and so on,
which is beyond my ability to go through.

627
00:30:13,720 --> 00:30:14,880
Okay, good.

628
00:30:14,880 --> 00:30:15,480
So far, so good.

629
00:30:15,480 --> 00:30:18,480
So we've done
two uses of pi one I guess one.

630
00:30:18,480 --> 00:30:20,880
So I don't know
how am I doing in time because I may,

631
00:30:22,120 --> 00:30:23,440
have five minutes.

632
00:30:23,440 --> 00:30:26,440
I'm nowhere close to what was it?

633
00:30:27,560 --> 00:30:29,960
Oh, 20 minutes.

634
00:30:29,960 --> 00:30:32,000
Okay. Oh, good. Then I'm very good.

635
00:30:32,000 --> 00:30:34,520
But I really panicked.

636
00:30:34,520 --> 00:30:38,120
I okay, I didn't practice,
so I have no idea how this how

637
00:30:38,120 --> 00:30:39,720
this is going to work, but.

638
00:30:39,720 --> 00:30:43,320
All right,
so let's go up and use the next homotopy.

639
00:30:44,200 --> 00:30:44,960
Right.

640
00:30:44,960 --> 00:30:48,880
So the next one is we're going to use maps
from a

641
00:30:49,440 --> 00:30:52,240
two sphere,
the usual sphere that we introduced.

642
00:30:52,240 --> 00:30:55,240
And we're going to map it
into the space of interest to us

643
00:30:55,560 --> 00:30:58,560
okay. So.

644
00:30:59,640 --> 00:31:02,280
We will make the other space be a sphere.

645
00:31:02,280 --> 00:31:03,600
And you'll see how.

646
00:31:03,600 --> 00:31:07,840
So we are interested in the second
homotopy group of the sphere.

647
00:31:07,840 --> 00:31:10,840
So maps from the sphere to the sphere.

648
00:31:10,840 --> 00:31:14,160
And it turns out
that's also the group of integers,

649
00:31:14,920 --> 00:31:17,640
which by the way
also means that the group is abelian.

650
00:31:17,640 --> 00:31:19,480
You know, you can add them in
whatever order.

651
00:31:19,480 --> 00:31:21,400
This is not always true of homotopy
groups.

652
00:31:21,400 --> 00:31:23,280
You can certainly come up with ones
which are

653
00:31:23,280 --> 00:31:26,080
which are non-abelian,
where the order doesn't matter.

654
00:31:26,080 --> 00:31:27,000
Okay.

655
00:31:27,000 --> 00:31:27,360
All right.

656
00:31:27,360 --> 00:31:29,240
So the index is going to be
the number of times

657
00:31:29,240 --> 00:31:32,000
the first sphere wraps around
the second sphere. So let's take a look.

658
00:31:32,000 --> 00:31:34,640
So here's again the trivial map.

659
00:31:34,640 --> 00:31:38,760
So the trivial map consists
of simply mapping all points.

660
00:31:38,760 --> 00:31:39,640
What do I do with this.

661
00:31:41,440 --> 00:31:45,200
Oh all points on the first sphere
which is the test sphere

662
00:31:45,600 --> 00:31:48,640
onto exactly the same point,
which of course I can pick to be anything.

663
00:31:49,480 --> 00:31:51,840
And as you can see, the entire area

664
00:31:51,840 --> 00:31:55,680
of the first sphere
has been mapped onto zero area.

665
00:31:55,680 --> 00:31:57,720
On the second sphere.

666
00:31:57,720 --> 00:31:59,880
By the way, you know how I produce this.

667
00:31:59,880 --> 00:32:01,600
I'm absolutely hopeless.

668
00:32:01,600 --> 00:32:04,560
I went into ChatGPT

669
00:32:04,560 --> 00:32:07,560
and said,
write me a piece of Mathematica code

670
00:32:07,680 --> 00:32:10,680
to do x spattered Mathematica code,

671
00:32:11,160 --> 00:32:14,160
put it into Mathematica, produce
this right?

672
00:32:15,720 --> 00:32:17,160
I did try earlier to say

673
00:32:17,160 --> 00:32:20,160
I'm giving a talk tomorrow.

674
00:32:21,720 --> 00:32:22,720
On the following topic.

675
00:32:22,720 --> 00:32:25,000
And it said in its usual cheery fashion,
sure.

676
00:32:25,000 --> 00:32:26,920
You know, what would you like me to do?

677
00:32:26,920 --> 00:32:30,560
So I dictated, you know, a couple
of paragraphs and it produced slides.

678
00:32:30,760 --> 00:32:34,200
In fact, I'm using the PowerPoint slides
that it started with.

679
00:32:34,200 --> 00:32:35,880
Of course
I had to because, you know, it wasn't.

680
00:32:36,960 --> 00:32:38,240
It will be by next January.

681
00:32:38,240 --> 00:32:41,240
But this and I'm not even kidding.

682
00:32:41,640 --> 00:32:44,400
But it was it produced a basic structure.

683
00:32:44,400 --> 00:32:47,760
It understood words this that's on
and so forth, you know, and kind of stuff.

684
00:32:47,760 --> 00:32:52,120
So we're,
as I said, a subject for a different talk.

685
00:32:53,720 --> 00:32:55,880
Which I don't know
if some of you may have noticed, I,

686
00:32:55,880 --> 00:32:57,760
I ran a series on AI these days.

687
00:32:57,760 --> 00:32:59,280
The first one was the Dominic Cummings.

688
00:32:59,280 --> 00:33:01,120
Josh Simons is coming next.

689
00:33:01,120 --> 00:33:04,200
And, so that's the parallel track
of trying to keep track of

690
00:33:04,200 --> 00:33:06,960
what's going on in the world.
That's it. That's how I produced this.

691
00:33:06,960 --> 00:33:08,040
But this is a trivial map.

692
00:33:08,040 --> 00:33:10,480
This entire area gets mapped on to zero.

693
00:33:10,480 --> 00:33:11,880
The next map.

694
00:33:11,880 --> 00:33:13,920
Much as for the circle one,

695
00:33:13,920 --> 00:33:16,800
is this one
in which every point in the first sphere

696
00:33:16,800 --> 00:33:19,440
gets mapped to the corresponding point
in the second sphere.

697
00:33:19,440 --> 00:33:20,880
And you can think of this
I took, you know, and

698
00:33:20,880 --> 00:33:24,360
basically I've wrapped the first sphere,
you know, just around that.

699
00:33:24,640 --> 00:33:29,240
And so the index is, you know, it's Iraq.

700
00:33:29,240 --> 00:33:32,360
Once every little area on the first year
went to the corresponding

701
00:33:32,360 --> 00:33:34,800
little area on the second one, and so on.

702
00:33:37,000 --> 00:33:38,160
How do I get the next one?

703
00:33:38,160 --> 00:33:41,160
Well,
so here's a picture stolen from Wikipedia.

704
00:33:42,360 --> 00:33:44,520
So okay, what's the picture?

705
00:33:44,520 --> 00:33:48,360
It's really you should these two halves
have to be glued together?

706
00:33:48,360 --> 00:33:50,040
They've been cut open to show you
what's going on.

707
00:33:50,040 --> 00:33:53,800
So you took the first sphere
and you started wrapping it around the

708
00:33:53,800 --> 00:33:54,360
second sphere.

709
00:33:54,360 --> 00:33:57,440
But you go around twice
before you, before you're done.

710
00:33:57,920 --> 00:34:00,680
So if you go, it was like with the vertex.

711
00:34:00,680 --> 00:34:02,920
You went once around, right?

712
00:34:02,920 --> 00:34:04,320
And the spin rotated twice.

713
00:34:04,320 --> 00:34:06,280
So in this case
you go first around the first sphere

714
00:34:06,280 --> 00:34:09,280
and you end up going twice
around the second sphere on the equator.

715
00:34:09,320 --> 00:34:11,560
And in this case
you've chosen a particular axis.

716
00:34:11,560 --> 00:34:14,640
But of course, if you took the whole thing
and rotated this way, that would be

717
00:34:14,640 --> 00:34:17,640
a continuous deformation and that would be
a map of the same degree.

718
00:34:18,000 --> 00:34:18,440
Right?

719
00:34:18,440 --> 00:34:22,000
So maps of the sphere to the sphere
have this character,

720
00:34:22,480 --> 00:34:26,400
and that's the second home
which pick up speed off the sphere.

721
00:34:26,920 --> 00:34:28,520
So so far so good.

722
00:34:28,520 --> 00:34:30,120
So we're up to pi two of S2.

723
00:34:30,120 --> 00:34:31,800
Now the question
is what can we use it for.

724
00:34:33,200 --> 00:34:35,720
So we're going to use it for something

725
00:34:35,720 --> 00:34:38,280
which is close to what we did

726
00:34:38,280 --> 00:34:41,280
with the xy vertex.

727
00:34:41,360 --> 00:34:43,680
Except it won't be a defect.

728
00:34:43,680 --> 00:34:45,640
It will be a solid term.

729
00:34:45,640 --> 00:34:48,560
So what's the difference
between these x and solid turns.

730
00:34:48,560 --> 00:34:51,800
So with the defect, as the term suggests,
there's something wrong with it.

731
00:34:52,440 --> 00:34:55,440
And what's wrong with
it is the region in the center

732
00:34:55,680 --> 00:34:58,720
where, you know, spins are really moving

733
00:34:58,720 --> 00:35:01,720
in a way that really wouldn't like to,

734
00:35:02,760 --> 00:35:03,120
because

735
00:35:03,120 --> 00:35:06,360
energetically it's very costly,
but far from the defect.

736
00:35:07,000 --> 00:35:07,320
You know,

737
00:35:07,320 --> 00:35:10,400
they're they're behaving in the way
that they would close to the ground state.

738
00:35:11,400 --> 00:35:14,240
Now, the reason defects are important is,

739
00:35:14,240 --> 00:35:17,880
is, is the topology,
which is if you have one, it's stuck.

740
00:35:17,880 --> 00:35:19,640
You can't easily get it out.

741
00:35:19,640 --> 00:35:23,600
So even with solid pieces of metal, right,
which are so on and so forth,

742
00:35:24,640 --> 00:35:26,600
you know, there's holes

743
00:35:26,600 --> 00:35:30,520
when you bend them and you have trouble,
you know, bending a piece of metal back.

744
00:35:30,520 --> 00:35:33,520
It's because you've introduced defects
which are then hard to get rid of.

745
00:35:34,200 --> 00:35:37,200
So, so, so defects are very,
very important because,

746
00:35:37,560 --> 00:35:38,160
you know, they're there.

747
00:35:38,160 --> 00:35:40,320
You can't easily get rid of them.

748
00:35:40,320 --> 00:35:42,440
Solid towns are not defects.

749
00:35:42,440 --> 00:35:45,440
They have often similar sort of,

750
00:35:46,240 --> 00:35:46,840
properties.

751
00:35:46,840 --> 00:35:48,840
And we'll use the same topology.

752
00:35:48,840 --> 00:35:51,840
But things are everywhere smooth.

753
00:35:52,120 --> 00:35:55,600
So in this case we're going to use pi
two of S2 to produce

754
00:35:56,160 --> 00:35:59,160
a solid on a topologically solid turn.

755
00:35:59,960 --> 00:36:04,360
And they've come to be called squirmy arms
and condensed matter physics.

756
00:36:04,880 --> 00:36:07,880
They were actually nicknamed babies
fermions,

757
00:36:08,360 --> 00:36:11,360
because the actual skyrmions, due
to, Tony's come.

758
00:36:11,560 --> 00:36:13,360
I can't work not very far from here.

759
00:36:14,640 --> 00:36:15,240
We're actually

760
00:36:15,240 --> 00:36:18,400
meant to be in one more dimension
and with,

761
00:36:19,080 --> 00:36:23,000
you know, a bigger sort of space of spins

762
00:36:23,280 --> 00:36:26,280
because he was thinking of how to get.

763
00:36:29,600 --> 00:36:31,680
Neutrons and protons, nucleons.

764
00:36:31,680 --> 00:36:35,000
Starting from a field theory of pions.

765
00:36:36,920 --> 00:36:38,720
There's no point in getting into
why he was trying to do it.

766
00:36:38,720 --> 00:36:40,160
It was a brilliant idea.

767
00:36:40,160 --> 00:36:43,800
It was ultimately completed by and with me
because the big challenge was,

768
00:36:43,800 --> 00:36:45,200
how do you get Fermi statistics

769
00:36:45,200 --> 00:36:48,200
from things that you start off with,
which are which are Bosnich.

770
00:36:48,960 --> 00:36:51,600
But anyway, the, the hence the term scum

771
00:36:51,600 --> 00:36:54,600
want so.

772
00:36:56,000 --> 00:36:58,360
These fermions had the following property.

773
00:36:58,360 --> 00:36:59,040
They.

774
00:36:59,040 --> 00:37:02,440
So we need two spheres
right to map between.

775
00:37:02,440 --> 00:37:05,440
So the first sphere will be

776
00:37:05,840 --> 00:37:08,760
the space on which the soliton lives.

777
00:37:08,760 --> 00:37:11,840
And the second is going to be
the space of fixed

778
00:37:11,840 --> 00:37:16,080
length spins, which unlike Z spins
now will be spins in three dimensions.

779
00:37:16,080 --> 00:37:19,680
So fixed length vectors in three
dimensions live on a sphere, right?

780
00:37:19,920 --> 00:37:24,040
So you're going to map from
a spatial sphere sphere to a spin sphere

781
00:37:24,920 --> 00:37:25,600
okay.

782
00:37:25,600 --> 00:37:28,720
So this is going to appear
in two dimensional space systems

783
00:37:28,720 --> 00:37:29,760
and field theories.

784
00:37:29,760 --> 00:37:32,280
But now let me walk you through
how that works okay.

785
00:37:32,280 --> 00:37:33,040
So first thing.

786
00:37:35,880 --> 00:37:36,160
Let's

787
00:37:36,160 --> 00:37:39,160
assume
that the space on which the spins live

788
00:37:39,360 --> 00:37:41,920
is literally a sphere, right.

789
00:37:41,920 --> 00:37:43,640
That's not going to be true in a solid.

790
00:37:43,640 --> 00:37:46,640
But let's let's start with that.

791
00:37:46,800 --> 00:37:48,760
Yeah.

792
00:37:48,760 --> 00:37:51,760
By the way,
same deals ChatGPT Mathematica.

793
00:37:55,000 --> 00:37:58,000
So here's the ground state

794
00:37:58,080 --> 00:37:59,600
spins agree everywhere.

795
00:37:59,600 --> 00:38:01,560
Right. Again,
I picked an arbitrary direction.

796
00:38:01,560 --> 00:38:03,880
It could have been
I could rotate all the spins

797
00:38:03,880 --> 00:38:05,560
in some direction
and that would be just as fine.

798
00:38:05,560 --> 00:38:08,680
So this is I've decorated, as it were,
the spatial sphere

799
00:38:09,000 --> 00:38:12,000
with this,

800
00:38:12,120 --> 00:38:14,760
Okay, then

801
00:38:14,760 --> 00:38:16,640
this is

802
00:38:16,640 --> 00:38:18,600
this isn't quite right.

803
00:38:18,600 --> 00:38:21,800
And so just ignore that the way
you're supposed to think about it.

804
00:38:21,800 --> 00:38:25,320
Focus on this patch is that at each point
in the spatial sphere,

805
00:38:25,720 --> 00:38:28,080
the spin points radially outwards.

806
00:38:28,080 --> 00:38:30,960
So it's like a hedgehog perfect hedgehog.

807
00:38:30,960 --> 00:38:32,200
Right? Okay.

808
00:38:32,200 --> 00:38:35,080
So this is a map

809
00:38:36,200 --> 00:38:38,520
in which each point on the spatial sphere

810
00:38:38,520 --> 00:38:41,720
has been mapped to the corresponding
point on the spin sphere.

811
00:38:41,720 --> 00:38:45,840
This is topologically distinct
small deformations of it I could rotate,

812
00:38:45,840 --> 00:38:47,560
you know, nearby spins a little bit
this way.

813
00:38:47,560 --> 00:38:51,000
That way won't change the fact
that it's very far from the ground state,

814
00:38:51,000 --> 00:38:53,200
but everybody with points
pointing the same.

815
00:38:53,200 --> 00:38:54,560
So far, so good.

816
00:38:54,560 --> 00:38:56,400
So we should agree.

817
00:38:56,400 --> 00:38:57,640
Now this is not a defect.

818
00:38:57,640 --> 00:38:59,720
Everything is smooth
as you walk along the sphere.

819
00:38:59,720 --> 00:38:59,920
Right.

820
00:38:59,920 --> 00:39:02,040
There's there's not a point
at which you see that anything

821
00:39:02,040 --> 00:39:05,440
particularly violent,
this is just slowly rotating.

822
00:39:06,240 --> 00:39:07,680
So this is a okay.

823
00:39:07,680 --> 00:39:12,280
Now the next thing I want to show you
is that we don't have to have

824
00:39:12,280 --> 00:39:15,880
the system live literally on a sphere
in order for this to, to work,

825
00:39:16,120 --> 00:39:20,800
we can actually descend to the plane,
which is where you can find systems.

826
00:39:20,800 --> 00:39:23,800
And the way to do that is stereographic
projection.

827
00:39:24,000 --> 00:39:24,600
Right.

828
00:39:24,600 --> 00:39:27,720
So stereographic
projection is going to tell us that,

829
00:39:29,480 --> 00:39:31,920
you know, corresponding
points on the sphere will be mapped

830
00:39:31,920 --> 00:39:36,360
to points on the plane, and the north pole
will eventually be mapped to infinity.

831
00:39:36,840 --> 00:39:37,560
Right.

832
00:39:37,560 --> 00:39:41,400
But the thing we need on the
when we do the stereographic projection,

833
00:39:41,760 --> 00:39:46,640
whatever the spin was doing at the North
Pole, will be what this spins

834
00:39:46,640 --> 00:39:49,640
will be doing anywhere at the boundary
of an extremely large system.

835
00:39:50,000 --> 00:39:52,080
So in all directions
you should get the same answer.

836
00:39:52,080 --> 00:39:52,520
And if you have

837
00:39:52,520 --> 00:39:56,360
such configurations on the plane,
you can come back to fly the plane,

838
00:39:56,360 --> 00:39:59,360
which is to say reverse started
vertically projected to the sphere.

839
00:39:59,520 --> 00:40:02,560
And so then you can use the topology
which was so manifest on the sphere.

840
00:40:03,560 --> 00:40:05,440
So here's the ground state.

841
00:40:05,440 --> 00:40:07,080
Everybody points the same way.

842
00:40:07,080 --> 00:40:09,120
Go off to infinity.
Everybody points the same way.

843
00:40:10,680 --> 00:40:11,320
The next map

844
00:40:11,320 --> 00:40:15,080
remember was more or less
right from the each point.

845
00:40:15,280 --> 00:40:17,440
So the North Pole
was going to go to the North Pole.

846
00:40:17,440 --> 00:40:20,320
The South Pole
was going to go to the South Pole.

847
00:40:20,320 --> 00:40:23,400
So here's the fermion drawn on the plane.

848
00:40:24,360 --> 00:40:26,360
So the South Pole was
the point of the origin.

849
00:40:26,360 --> 00:40:28,800
It's not perfect,
but this thing is pointing down.

850
00:40:28,800 --> 00:40:29,520
And as you can see,

851
00:40:29,520 --> 00:40:33,440
as you go to infinity along any direction,
the spin goes from being down

852
00:40:34,000 --> 00:40:38,360
to being smoothly up and everywhere else,
almost up, everywhere at the boundary.

853
00:40:39,120 --> 00:40:41,640
Spins are pretty much recovered
to being up.

854
00:40:41,640 --> 00:40:45,720
So in a magnet in the lab

855
00:40:46,800 --> 00:40:49,800
or some system with this kind of,

856
00:40:49,920 --> 00:40:52,760
order parameter, what you should then

857
00:40:52,760 --> 00:40:55,760
expect to find are

858
00:40:56,480 --> 00:41:00,080
excitations of the ground state,
which locally have this structure.

859
00:41:00,080 --> 00:41:03,520
They look like particles because
after a while, you know, they recover.

860
00:41:04,000 --> 00:41:06,960
And these particles are called skyrmions.

861
00:41:06,960 --> 00:41:08,160
So for example,

862
00:41:10,760 --> 00:41:11,560
probably

863
00:41:11,560 --> 00:41:15,400
which most completely,
you know, sort of does what you want to do

864
00:41:15,400 --> 00:41:18,800
is actually in the quantum Hall effect,
something that was my thesis.

865
00:41:19,640 --> 00:41:22,640
And you get these particles,
they actually have,

866
00:41:23,080 --> 00:41:26,600
charge, which in the case of the one third
Hall stage is the fractional charge

867
00:41:26,600 --> 00:41:27,600
one third of an electron.

868
00:41:27,600 --> 00:41:28,720
They have a funny statistic.

869
00:41:28,720 --> 00:41:31,720
So they really I think they would
have made Tony Scott very happy.

870
00:41:32,120 --> 00:41:32,760
They do everything.

871
00:41:32,760 --> 00:41:34,640
So they're genuinely new
kinds of particles.

872
00:41:34,640 --> 00:41:36,720
But with this kind of description.

873
00:41:36,720 --> 00:41:39,240
Okay. So that came out.

874
00:41:39,240 --> 00:41:43,080
Now again I said, you know,
topology proposes an energetic disposes.

875
00:41:43,080 --> 00:41:46,080
In this case, it turns out that
the size of this curve means

876
00:41:46,080 --> 00:41:48,680
is itself actually can vary.

877
00:41:48,680 --> 00:41:49,880
They're not like this.

878
00:41:49,880 --> 00:41:52,400
And the particular energetics of systems
is important.

879
00:41:52,400 --> 00:41:54,960
And the quantum Hall effect,
they end up having a nice healthy size,

880
00:41:54,960 --> 00:41:57,960
which actually if you do in parameters
you can make as big as you want.

881
00:41:58,040 --> 00:41:59,480
And then they are

882
00:41:59,480 --> 00:42:02,760
nice objects and lots of their physics
can be understood that way.

883
00:42:03,720 --> 00:42:05,880
All right.

884
00:42:05,880 --> 00:42:08,880
Good. So,

885
00:42:08,920 --> 00:42:11,920
next up and the last,

886
00:42:12,240 --> 00:42:14,040
thing I'll do

887
00:42:14,040 --> 00:42:17,040
is to use the same topology, Pi two of S2,

888
00:42:17,480 --> 00:42:21,000
to tell you something
about how one gets fancy band insulators.

889
00:42:21,840 --> 00:42:23,960
So this is,

890
00:42:23,960 --> 00:42:25,480
so solid state 101.

891
00:42:25,480 --> 00:42:28,040
Just to remind you where bands come from,
you start with, let's say,

892
00:42:28,040 --> 00:42:30,920
the dispersion of a free particle,
which is a parabola,

893
00:42:30,920 --> 00:42:33,600
and then you put on a weak
periodic potential

894
00:42:33,600 --> 00:42:36,200
and that has the annoying habit

895
00:42:36,200 --> 00:42:39,520
of causing things to mix across a certain

896
00:42:39,960 --> 00:42:42,920
right momentum range,
which is the inverse of the distance scale

897
00:42:42,920 --> 00:42:45,360
that you've introduced.
And that causes gaps to open.

898
00:42:45,360 --> 00:42:48,120
And then you fold them
and draw them this way,

899
00:42:48,120 --> 00:42:51,280
because once you don't have perfect
translation invariance and you want

900
00:42:51,280 --> 00:42:54,280
only translation invariant
up to a certain distance,

901
00:42:54,360 --> 00:42:56,160
legally,
there's no reason for you to insist

902
00:42:56,160 --> 00:42:57,440
that there's something called a momentum.

903
00:42:57,440 --> 00:42:59,760
Momentum as a consequence
of translation invariance.

904
00:42:59,760 --> 00:43:03,600
And if translation invariance
is only do there up to a certain,

905
00:43:03,960 --> 00:43:07,720
you know, periodicity, then momentum is
only defined up with that periodicity.

906
00:43:08,040 --> 00:43:10,400
And there's no meaning
to drawing this separate from that.

907
00:43:10,400 --> 00:43:11,640
So you might as well draw it this way.

908
00:43:12,720 --> 00:43:15,840
Once you do that, what you have
is that you have a set of energy bands.

909
00:43:15,840 --> 00:43:19,320
Each one of these and the energy bands
are a function of momentum, but

910
00:43:19,960 --> 00:43:22,960
they have to be periodic. So

911
00:43:23,760 --> 00:43:26,080
and then you go up in higher dimensions,
they have to be periodic

912
00:43:26,080 --> 00:43:27,840
in every direction of momentum.

913
00:43:27,840 --> 00:43:31,200
So if you take the underlying space here,
which is momentum space

914
00:43:31,200 --> 00:43:33,840
and you insist the things on it
have to be periodic,

915
00:43:33,840 --> 00:43:37,640
it says if the momentum
lives on a torus like this,

916
00:43:38,120 --> 00:43:42,480
so the Brillouin zones of solids
are to try

917
00:43:42,760 --> 00:43:46,240
in one day is just a circle to the it's
a more familiar Taurus.

918
00:43:46,240 --> 00:43:47,640
And so

919
00:43:47,640 --> 00:43:50,400
okay, good. Now

920
00:43:50,400 --> 00:43:54,120
this is one day which is very special
and kind of not that interesting.

921
00:43:54,120 --> 00:43:56,760
But once you move up from one dimension,
interesting things can happen.

922
00:43:56,760 --> 00:44:00,720
So for example, you can have
the two of these bands are sort of there,

923
00:44:00,720 --> 00:44:03,720
and I can deform the Hamiltonian
and do things to it.

924
00:44:04,000 --> 00:44:07,000
So that states in here
mix among each other.

925
00:44:07,360 --> 00:44:10,160
Maybe at some stage
the gap closes and it opens again.

926
00:44:10,160 --> 00:44:12,200
Well,
I don't really affect stuff that happens.

927
00:44:12,200 --> 00:44:13,760
You know, further up.

928
00:44:13,760 --> 00:44:16,000
So for this reason, it's actually

929
00:44:16,000 --> 00:44:19,800
perfectly legitimate
to think about what happens

930
00:44:19,800 --> 00:44:24,680
when you have a finite number
of bands living on atoms.

931
00:44:26,040 --> 00:44:26,280
Okay.

932
00:44:26,280 --> 00:44:29,280
So this is
this is when we get to that mathematics.

933
00:44:30,720 --> 00:44:33,280
And then the okay.

934
00:44:33,280 --> 00:44:36,320
So so that's
the next step is for instance,

935
00:44:36,320 --> 00:44:40,320
we could consider two dimensions
and we could say just take two bands.

936
00:44:40,880 --> 00:44:44,160
And the thing I'm not telling you
is that the way that the Hamiltonian of

937
00:44:44,160 --> 00:44:48,160
interest will always was for single
this a single particle physics independent

938
00:44:48,160 --> 00:44:51,960
particle physics will will factor
at each point in momentum space.

939
00:44:52,120 --> 00:44:52,720
Right.

940
00:44:52,720 --> 00:44:55,560
So so because of that,

941
00:44:55,560 --> 00:44:58,560
what this formula is telling you is

942
00:44:58,960 --> 00:45:00,160
considered a given point.

943
00:45:00,160 --> 00:45:01,480
Okay.

944
00:45:01,480 --> 00:45:04,480
Consider
a three dimensional vector of unit length

945
00:45:04,920 --> 00:45:08,560
and consider the set of poly matrices
sigma x, sigma y and sigma z.

946
00:45:09,120 --> 00:45:12,400
And then this dot product
gives you the most what is all it's

947
00:45:12,400 --> 00:45:15,720
really telling you is the most general
Hermitian two by two matrix.

948
00:45:15,720 --> 00:45:18,400
You can write
can be written like this. That's it.

949
00:45:18,400 --> 00:45:21,520
And the Hamiltonian has to be here
at each point in the language that.

950
00:45:24,000 --> 00:45:24,320
Okay.

951
00:45:24,320 --> 00:45:27,000
So what have I done?

952
00:45:27,000 --> 00:45:29,640
I'm telling you
that the way to think about a two band

953
00:45:29,640 --> 00:45:33,200
problem is to think in terms
of these vectors

954
00:45:33,840 --> 00:45:36,840
and of k, the vectors live on this sphere,

955
00:45:37,440 --> 00:45:40,280
but k lives on the tourist.

956
00:45:40,280 --> 00:45:43,320
So at this point I should go through

957
00:45:43,760 --> 00:45:46,440
a lot of painful topology,

958
00:45:46,440 --> 00:45:47,680
to work that out.

959
00:45:47,680 --> 00:45:51,320
But actually take my word for it, I can
replace d2 by S2 and get the same answer.

960
00:45:51,320 --> 00:45:53,160
It's not obvious, right?

961
00:45:53,160 --> 00:45:55,320
But but it's true.

962
00:45:55,320 --> 00:45:59,040
So with that, I'm in business
because we've already discussed

963
00:45:59,040 --> 00:46:02,560
what happens for maps between S2 and S2,

964
00:46:03,840 --> 00:46:05,400
so it's sort of funny.

965
00:46:05,400 --> 00:46:10,680
I'm applying topology
now to classify these band Hamiltonians.

966
00:46:11,040 --> 00:46:14,200
I'm saying if there's a two by two,
if I have two bands,

967
00:46:14,960 --> 00:46:19,080
any single particle Hamiltonian
independent electrons not interacting

968
00:46:19,080 --> 00:46:25,480
electrons, I put in it will be classified
by some map of the sphere to the sphere.

969
00:46:25,480 --> 00:46:27,560
So there must be an integers
worth of them.

970
00:46:27,560 --> 00:46:30,080
There is some trivial map
where all the ends are the same.

971
00:46:30,080 --> 00:46:31,320
Maybe point in the z direction.

972
00:46:31,320 --> 00:46:33,240
It looks like the polling matrix
everywhere,

973
00:46:33,240 --> 00:46:36,000
and otherwise
there will be things that were win,

974
00:46:36,000 --> 00:46:39,480
and in fact these ends will do
pretty much what those fermions were doing

975
00:46:40,200 --> 00:46:43,200
that are,
you know, had in the in the last picture.

976
00:46:43,360 --> 00:46:44,400
Okay.

977
00:46:44,400 --> 00:46:48,680
So the subject of topological insulators
precedes this way.

978
00:46:49,120 --> 00:46:50,360
This is the simplest case.

979
00:46:51,320 --> 00:46:54,320
You use somewhat
fancier version of this called k theory.

980
00:46:54,320 --> 00:46:56,560
Then you have to worry about disorder
and so on and so forth.

981
00:46:56,560 --> 00:47:00,440
But basically you start from here
and you make your way up the,

982
00:47:01,440 --> 00:47:04,200
hierarchy of powerful topological tools.

983
00:47:04,200 --> 00:47:07,200
Now you might say, okay, this is true.

984
00:47:07,640 --> 00:47:08,640
Why do I care?

985
00:47:08,640 --> 00:47:10,200
Does this have anything to do
with anything?

986
00:47:10,200 --> 00:47:10,520
Right.

987
00:47:10,520 --> 00:47:13,080
That's an utterly reasonable question.

988
00:47:13,080 --> 00:47:15,480
And the answer is, well,

989
00:47:15,480 --> 00:47:20,280
if you have a nontrivial winding
of the Hamiltonian, it has two bands.

990
00:47:20,680 --> 00:47:23,400
If you fill one of them with electrons,

991
00:47:23,400 --> 00:47:26,400
yeah, you're going to find
that the whole conductance.

992
00:47:27,200 --> 00:47:31,080
So electric field this way
current off of it right is actually going

993
00:47:31,080 --> 00:47:32,280
to end up being given by the.

994
00:47:32,280 --> 00:47:35,120
So the winding number of times
E squared over

995
00:47:35,120 --> 00:47:36,720
the longitudinal conductance will be zero.

996
00:47:36,720 --> 00:47:39,720
Because there's a you know,
there's a gap in the energy spectrum.

997
00:47:40,320 --> 00:47:43,320
And so these things end up
exhibiting the quantum Hall effect

998
00:47:43,960 --> 00:47:48,600
in a lattice, which is not how it was
initially, you know, discovered and

999
00:47:49,920 --> 00:47:52,160
they are now called Shannon switches.

1000
00:47:52,160 --> 00:47:52,800
Right.

1001
00:47:52,800 --> 00:47:56,840
So the same topology allowed us
to discover that

1002
00:47:57,720 --> 00:48:00,040
these Chern insulators can exist with

1003
00:48:00,040 --> 00:48:04,520
quantized Hall conductance one to
and then plus minus one, you know,

1004
00:48:04,800 --> 00:48:07,800
plus minus one, minus two,
one time, which has to do with

1005
00:48:08,520 --> 00:48:11,800
you literally get the opposite sign
of the Hall effect in a in a given sample.

1006
00:48:12,960 --> 00:48:15,840
All right.

1007
00:48:15,840 --> 00:48:18,840
So I think.

1008
00:48:20,520 --> 00:48:20,760
Okay.

1009
00:48:20,760 --> 00:48:24,800
So I'll just mention this for those of you
who are a glutton for punishment,

1010
00:48:24,800 --> 00:48:26,400
which is that

1011
00:48:26,400 --> 00:48:28,680
I talked about the Hamiltonian,
but actually,

1012
00:48:28,680 --> 00:48:30,200
you know, I only failed one band.

1013
00:48:30,200 --> 00:48:31,760
So somebody may be saying, well,

1014
00:48:31,760 --> 00:48:33,120
I don't have to think
about the Hamiltonian.

1015
00:48:33,120 --> 00:48:35,920
Why don't I just think
about the actual state, the wave function.

1016
00:48:35,920 --> 00:48:40,000
You can and you can get the same answer
that requires a different fiber bundle.

1017
00:48:40,000 --> 00:48:42,040
And you know,
that would require a little more work.

1018
00:48:43,240 --> 00:48:43,720
Okay.

1019
00:48:43,720 --> 00:48:44,640
So I'm done.

1020
00:48:44,640 --> 00:48:46,320
So there are many uses of topology.

1021
00:48:46,320 --> 00:48:48,120
Right.

1022
00:48:48,120 --> 00:48:51,920
All the spaces we talked about
and we explored a few defects,

1023
00:48:51,920 --> 00:48:54,920
all of forms, topological terms
and band theory.

1024
00:48:55,440 --> 00:48:58,400
And then, you know,
there's been this huge try to work.

1025
00:48:58,400 --> 00:49:00,640
As is often the case, these days.

1026
00:49:00,640 --> 00:49:04,200
It's not of theory tends to outrun
experiments, which are much harder to do,

1027
00:49:04,600 --> 00:49:05,920
but nevertheless,

1028
00:49:05,920 --> 00:49:08,880
especially with regard to,
you know, single electron physics

1029
00:49:08,880 --> 00:49:11,520
that's actually been a remarkable amount
of experimental work

1030
00:49:11,520 --> 00:49:14,520
realizations, materials, clever
material physics,

1031
00:49:14,760 --> 00:49:17,400
most of which I haven't even
I tried to describe to you.

1032
00:49:17,400 --> 00:49:20,400
And with that, I will stop.

1033
00:49:28,560 --> 00:49:31,560
So for the purposes of recording,
I'm going to repeat the question.

1034
00:49:31,680 --> 00:49:32,200
Which is to.

1035
00:49:32,200 --> 00:49:34,680
So the question, as I said,
it makes a difference.

1036
00:49:34,680 --> 00:49:37,640
But I didn't explain
what what the difference is

1037
00:49:37,640 --> 00:49:40,520
when we add the topological term
to the quantum mechanics.

1038
00:49:40,520 --> 00:49:43,520
So this is of course easiest to say
in the language of passenger intervals.

1039
00:49:43,920 --> 00:49:46,280
But I could just as well do it
in terms of Hamiltonians.

1040
00:49:46,280 --> 00:49:48,600
The energy spectrum
will be different, right.

1041
00:49:48,600 --> 00:49:51,480
So you'll get,
you know, different eigenvalues.

1042
00:49:51,480 --> 00:49:53,880
But if you calculate the potential
value and said, you know,

1043
00:49:56,880 --> 00:49:58,560
okay, so this particular one has a certain

1044
00:49:58,560 --> 00:50:01,520
circular symmetry,
let's say the energy spectrum is different

1045
00:50:01,520 --> 00:50:03,320
which is you get different energy
eigenvalues.

1046
00:50:03,320 --> 00:50:07,800
So it's so in Hamiltonian language
you will end up doing something like

1047
00:50:07,800 --> 00:50:11,160
either
you will explicitly put a vector potential

1048
00:50:11,640 --> 00:50:14,640
of the flux in the middle
and keep periodic boundary conditions,

1049
00:50:15,120 --> 00:50:16,120
or you are forced

1050
00:50:16,120 --> 00:50:19,680
to go to boundary conditions
in which the wave function has to twist

1051
00:50:20,160 --> 00:50:22,320
and come back by a factor
E to the I alpha.

1052
00:50:22,320 --> 00:50:23,680
And that is another way of saying

1053
00:50:23,680 --> 00:50:26,680
why the eigenfunctions
will then come with different eigenvalues.

1054
00:50:27,520 --> 00:50:31,280
So in the case of QCD,
which we can describe better,

1055
00:50:31,920 --> 00:50:34,560
there's a topological term
which will then break symmetries

1056
00:50:34,560 --> 00:50:37,320
and you know, CP
violation, things like that.

1057
00:50:37,320 --> 00:50:40,320
And that really sort of a descendent of,

1058
00:50:40,560 --> 00:50:43,320
ascendant of this, simple, simple concern.

1059
00:50:43,320 --> 00:50:44,880
Okay.
So the question is what did I write here?

1060
00:50:44,880 --> 00:50:49,320
So the first Lagrangian is just one
half the mass times the velocity squared

1061
00:50:50,360 --> 00:50:51,000
zero square.

1062
00:50:51,000 --> 00:50:52,080
Yeah.

1063
00:50:52,080 --> 00:50:54,400
And then the action

1064
00:50:54,400 --> 00:50:57,400
was just that the and plus.

1065
00:50:58,200 --> 00:51:01,200
Well let me answer this way,
which is to say

1066
00:51:02,640 --> 00:51:04,960
there

1067
00:51:04,960 --> 00:51:08,040
there are field theories
that people study mathematically

1068
00:51:08,160 --> 00:51:11,160
which are topological quantum
field theories.

1069
00:51:11,320 --> 00:51:14,320
You know, Steve Stockwell,
I don't know if you talk about them,

1070
00:51:14,320 --> 00:51:17,520
but they will be background music,

1071
00:51:18,040 --> 00:51:21,200
or maybe invisible, inaudible
background music to what he's saying.

1072
00:51:22,000 --> 00:51:25,920
But those are field theories
in which that imaginary

1073
00:51:26,600 --> 00:51:30,960
quantum mechanical universes
in which, yes, everything is topology.

1074
00:51:30,960 --> 00:51:32,840
There's literally no notion of distance.

1075
00:51:32,840 --> 00:51:34,760
Right.

1076
00:51:34,760 --> 00:51:36,840
So people study,

1077
00:51:36,840 --> 00:51:41,320
I think there's some moral gravity
problems, ideological with that.

1078
00:51:42,240 --> 00:51:44,080
Well, it's not the world you live in.

1079
00:51:44,080 --> 00:51:46,000
It's just it's simply not the world
you live in. Right?

1080
00:51:46,000 --> 00:51:48,960
Because if you live in London
versus living in Edinburgh,

1081
00:51:48,960 --> 00:51:51,240
I mean, it's very clear
that it takes longer to get there.

1082
00:51:53,280 --> 00:51:56,280
So, so that's why which is to say,

1083
00:51:57,280 --> 00:52:00,280
that so it's, it's it's all I mean,

1084
00:52:01,040 --> 00:52:04,160
it's instructive
to study these simple cases.

1085
00:52:04,480 --> 00:52:07,480
And so to get the,

1086
00:52:08,040 --> 00:52:09,480
that's not so much fun.

1087
00:52:09,480 --> 00:52:14,120
It's so the topological quantum field
theory is it's actually spacetime to

1088
00:52:14,640 --> 00:52:15,280
topological.

1089
00:52:15,280 --> 00:52:17,880
So both are have no clear meaning.

1090
00:52:17,880 --> 00:52:21,640
The only thing you know is did you do this
or did you not do this right around

1091
00:52:21,640 --> 00:52:25,320
so so so that the world around us
is knocked off?

1092
00:52:25,320 --> 00:52:26,520
Illogical in that sense.

1093
00:52:26,520 --> 00:52:30,120
It's not invariant to arbitrary,
you know, changes of position,

1094
00:52:30,360 --> 00:52:33,160
but nevertheless,
I mean, you know, a lot of

1095
00:52:33,160 --> 00:52:36,280
very beautiful mathematical physics,
which has been very insightful, actually

1096
00:52:36,280 --> 00:52:39,280
has has gone into exactly the universe
you would like to,

1097
00:52:40,440 --> 00:52:44,080
consider I plead guilty,

1098
00:52:46,360 --> 00:52:48,960
I see the. Yes.

1099
00:52:48,960 --> 00:52:51,840
No, no. Absolutely,
absolutely. Absolutely.

1100
00:52:51,840 --> 00:52:52,160
William.

1101
00:52:52,160 --> 00:52:52,720
Not so much.

1102
00:52:52,720 --> 00:52:55,280
But yeah, I mean,
although even in helium, in cold atomic.

1103
00:52:55,280 --> 00:52:57,400
Sorry,
I'm supposed to repeat the question.

1104
00:52:57,400 --> 00:53:00,400
I didn't mention, laboratory examples
of topological defects.

1105
00:53:00,760 --> 00:53:02,280
And you're right.

1106
00:53:02,280 --> 00:53:05,760
Liquid crystals, you know,
they're beautiful imaging in called

1107
00:53:05,760 --> 00:53:08,960
atomic gases where they have image them,
you know, you can you can see them.

1108
00:53:09,840 --> 00:53:11,360
And you're absolutely right.

1109
00:53:11,360 --> 00:53:13,920
Topological defects, cosmic strings.

1110
00:53:13,920 --> 00:53:16,760
I think that's still up for discussion.

1111
00:53:16,760 --> 00:53:19,760
So, but absolutely,

1112
00:53:20,640 --> 00:53:23,320
it's it's absolutely correct.

1113
00:53:23,320 --> 00:53:26,880
So the statement is that a vertex
anti vertex pair is not,

1114
00:53:27,160 --> 00:53:30,720
from a topological point of view,
is not a stable configuration.

1115
00:53:30,960 --> 00:53:32,560
It could disappear.

1116
00:53:32,560 --> 00:53:35,080
But if there is an isolated vertex
sitting there

1117
00:53:35,080 --> 00:53:38,080
it would require you to do something
on the entire size of the system.

1118
00:53:38,400 --> 00:53:40,320
So topology

1119
00:53:41,800 --> 00:53:44,160
that was
the statement that topology proposes.

1120
00:53:44,160 --> 00:53:46,800
But of course the details of physics
and energetics still remain.

1121
00:53:46,800 --> 00:53:51,360
Sometimes, you know, it can happen
that you make a vertex anti vertex

1122
00:53:51,360 --> 00:53:55,280
and they get stuck for some reason
so that they're forever right.

1123
00:53:55,280 --> 00:53:59,120
And so from a physics viewpoint,
they're infinitely long lived,

1124
00:53:59,120 --> 00:54:02,120
even though topology would say
there's no reason for them to be there.

1125
00:54:02,160 --> 00:54:05,200
So topology is really insightful

1126
00:54:05,200 --> 00:54:08,440
because when you start to deal
with complicated order parameter spaces,

1127
00:54:09,040 --> 00:54:12,840
so liquid crystals, you know, you can have
a sphere with amphipods identified.

1128
00:54:12,840 --> 00:54:15,440
You know, that's not something
we're used to thinking in everyday life.

1129
00:54:15,440 --> 00:54:19,480
Then these formal mathematical results
really come into their into their own,

1130
00:54:19,480 --> 00:54:20,280
because it allows you

1131
00:54:20,280 --> 00:54:23,280
to identify defects that you wouldn't
have been able to that easily get

1132
00:54:23,320 --> 00:54:25,320
if you were just sitting
and trying to draw pictures.

1133
00:54:26,280 --> 00:54:26,800
I mean,

1134
00:54:26,800 --> 00:54:29,800
eventually you would,
but it's just easier to look at the book.

1135
00:54:33,400 --> 00:54:36,400
And go, right.

1136
00:54:36,440 --> 00:54:37,880
But that's actually.