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Let's go for.

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Super 820.

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Thank you very much.

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And thank you all for sticking around.

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I'm Dumitru.

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I'm an LP fellow here at Oxford.

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And today I'd like to talk to you
about something deceptively simple.

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What happens when electrons
slow down dramatically?

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And the title of my talk
is The Physics of Flat Electrons.

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And as you will see,
this idea sits at the crossroads of,

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band theory,
topology and strong correlations.

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And it leads us to
some of the most exotic phases

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of quantum matter known today.

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Before I start, however,
let me make a disclaimer.

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I understand that
the title is a bit of a clickbait.

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I'm not a flat electron ist.

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We haven't discovered
the new fundamental particle.

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Flat electrons aren't exotic entities
beyond the Standard Model.

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They're just the plane of the electron.

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Same charge, same spin.

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But they behave in very unusual ways
inside certain materials.

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Now, this xkcd comic here

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makes a playful jab at the scientific
hierarchy.

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Physicists at the top, of course,
but it misses something crucial.

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The idea that biology is just applied
chemistry and so on breaks down.

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Once you realize that the key player
is emergent behavior.

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As Phil Anderson shown here,
famously put it, more is different.

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When electrons are confined to nearly flat
bands, their interactions dominate

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and they give rise to collective phenomena
that are completely invisible

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at the microscopic one electron level.

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So yes, they're just electrons,
but in the right environment

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they become the seeds
of superconductivity, magnetism, charge,

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fractional ization, things that no
individual electron can do on its own.

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So to

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understand what it means
for electrons to be flat,

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we first need to talk about what it means
for them to be dispersive.

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So before diving into all the exotic
physics, start

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with a quick refresher of bang theory
how electrons move in a crystal,

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how energy bands emerge,
and what we mean by dispersion.

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This will set the stage for understanding
what it means to suppress that motion,

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and how that can fundamentally reshape
the body landscape.

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So let's start from the free
electron model.

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If we place an electron
in a perfectly uniform

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background, as shown here on the left,
there is no potential at all.

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Its dispersion is just a plain,
simple parabola, right?

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We know that the energy's
h squared k squared over two m right

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t squared over to him.

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That's what we see on the left.

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Now when we move to a crystalline solid

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we break the continuous translation
symmetry to discrete translation symmetry.

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Right. In a crystal.

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The momentum is only defined up
to a reciprocal lattice vector.

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In this case it's two pi over A
where a is the lattice constant.

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This essentially falls back the dispersion

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of the free electron
inside the so-called Brillouin zone.

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But of course,

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the actual crystal does have a potential
with a discrete periodicity.

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And after we add the periodic potential
as shown on the right hand side,

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we see gaps opening and

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an end bands electronic bands forming.

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So what do we get at the end
as a band structure?

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Electrons still propagate,
but now they do so

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with a modified dispersion
that reflects the periodic environment.

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This is the basis of the nearly free
electron model.

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Still delocalized
still momentum eigenstates,

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but dressed by the
periodicity of the lattice.

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Now let's switch perspective

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and instead of thinking about
three particles perturbed by a lattice,

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we can start from tightly bound electrons,
say in atomic orbitals.

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Here I'm showing s p
and the orbital levels.

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They're localized in discrete right.

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I'm showing them here on the left
hand side.

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The first step is to imagine copying
this atomic orbitals across the lattice.

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Now we have one orbital
per unit cell repeated periodically.

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That gives us a flat band structure.

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No hopping, no dispersion.

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We simply multiplied all these energy
levels and add the momentum to them.

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Then when we allow this orbitals
to hybridize,

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which means we allow the electrons to hop
between them, then each level broadens

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into a band with a bandwidth controlled
by the hopping amplitude.

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This is the so-called tight binding

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model is the polar opposite
of the nearly free electron case.

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Instead of starting from momentum
eigenstates, we now start

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from a real state from real space orbitals
and let them communicate.

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In the end, both routes
lead us to the same concept of bands,

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which are just continuous energy
levels, right?

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That the electrons occupy,
which are labeled by crystalline momentum.

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Now let's talk about what

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happens when we account
for electron electron interactions.

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Electrons,
of course, they repel one another.

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They carry the same charge.

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So we can't just take those bands
and fill them

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with electrons
as if they were non-interacting particles.

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Real electrons collide and they scatter
and they exchange energy and momentum.

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But remarkably for many metals,
at low enough temperature,

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the system behaves
as if it were non-interacting.

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And this is the magic of land.

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That was Fermi liquid theory,

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one of the great triumphs
of condensed matter physics.

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What Landauer postulated and later
articles of proved,

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is that even when electron electron
interactions are present,

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the system can still be described
in terms of long lived quasi particles.

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These are not bare electrons anymore.

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They're dressed by the interaction
with the rest of the electronic see,

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and they require modified parameters
such as modified mass and lifetime.

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When I was

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at an undergrad in Cambridge,
when I was an undergrad in Cambridge,

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someone explained to me

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the concept of quasi particles
using this exact same diagram,

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saying the real particle is a horse
and the quasi particle is a quasi horse,

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which at the time seemed just as helpful
as it sounds.

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But to their credit,

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it is a vivid way to see that
the thing that you care about,

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the thing that you see,
isn't the bare object, the bare electron

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is the bare object
plus a cloud of particle hole excitation.

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Microscopically, this whole theory,
this whole Landau Fermi liquid theory

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can be understood
using many body perturbation theory,

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but critically, that only works

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when the electrons are dispersive enough
such that the interactions stay in check.

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Once that fails, the Fermi
liquid theory breaks down.

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And that's a very good thing,
because otherwise condensed matter physics

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would be a very boring subject.

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So now let's take this

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logic and flip it on its head.

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What happens when we're trying

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to partially fill a band that is flat?

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In that case, the kinetic energy,
the dispersion becomes negligible.

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That means that this bandwidth w is much,
much smaller than the interaction scale.

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Sometimes it's even much smaller
than the energy gap.

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This is the regime where Landau's theory
completely breaks down.

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In a dispersive band,
you could treat the interactions

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as a small perturbation
on top of a dispersive band,

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but in a flat band there's no free motion.

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There is nothing to perturb against.

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Your unperturbed
Hamiltonian is zero, right?

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So in fact,
the situation here is reversed, right?

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Interactions dominate.

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Then kinetic energy becomes a small
perturbation if it even exists at all.

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So we can't start from a non-interacting
solution and perturb our way out.

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We're forced forced to tackle
the many body problem full on.

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Now these flat

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bands may feel like a hot topic now,
but I've actually played

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quite an important role
in condensed matter for decades.

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You could loosely trace
some of these ideas

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back to heavy
fermion materials in the 70s.

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These aren't flat band systems
in the strict sense,

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but they're often group in the same family
because they exhibit

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narrow bands and strong
electron correlation.

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The first truly flat bands in the
in the in the sense of the term

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of today, appeared when Landau levels
in the quantum Hall effect,

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both in the integer and fractional
one were discovered.

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These discovered
discoveries were nothing short

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of revolutionary,
because they each earned the Nobel Prize

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and showed that the interaction
driven physics in flat bands

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can lead to entirely new quantum phases.

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The modern flat band renaissance

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came with twisted
bilayer graphene in 2018.

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So if you have two layers of graphene
that many of you

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might remember this from, last term's
a morning of theoretical Physics.

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If you have two layers of graphene
and you rotate them at an angle

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which is close to the magic angle, 
then you can actually show that flat bands

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where interactions dominate emerge
and you have superconductivity,

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correlated insulators, and more.

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Today, this platform is evolving.

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We now have fractional Chern insulators,
which basically exhibits

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the fractional quantum Hall effect
without an applied magnetic field.

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But there's a question

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which hasn't been answered until recently.

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Can we get these flat bands

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in conversion
of crystal ordinary crystals?

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No twisting, no stacking, no more magic,
no magnetic field.

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This is what I'm going
to try to answer in my talk.

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So to really motivate
why flat bands are exciting,

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let's take a closer look at a system
that recently reignited the field.

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Right.
This is twisted bilayer of graphene.

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So you take two sheets of graphene.

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You rotate, rotate them relative
to one another at an angle of 1.1 degrees,

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and the resulting moiré patterns flattens
the bat,

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flattens
the bands near, charge neutrality.

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What you're seeing here on the left,
I actually eight

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nearly flat degenerate flat bands. Right.

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Two for each value, two for each valley,
two for each spin and two for each layer.

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They're clustered around the zero energy.

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Now here's the key.

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These gap, these black bands are gapless
in the non-interacting model.

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But if you try experimentally,
if you try to field this bands

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and you fill an integer number of them,
you see that the interaction effects

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kick in and the system develops
and gaps open.

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These are so-called correlated
insulator states labeled here

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in this schematic diagram
on the on the upper right.

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Between these feelings the system
can enter a superconducting phase.

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These are the blue SC domes
which emerge entirely from interactions.

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Since there is no phonon glue here,

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and the richness doesn't stop here
actually.

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This is bilayer

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graphene has shown signatures
of something called the Pomerantz effect,

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which means that

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with increasing temperature,
you actually cause the system to order.

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This is somewhat, analogous
to the permanent formation shock effect

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in, in, liquid helium.

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We're actually heating the system, 
makes, makes liquid helium freeze.

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So you have only one material,
and the only knob is the carrier density.

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And we see correlated
insulator superconductors, strange

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metals, quantum dot behavior,
all stemming from this flat pass.

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Now, to really

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understand
the interplay between interactions

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and band structure,
let's start with the Hubbard model.

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The Hubbard model is arguably

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the simplest model
of interacting electrons on a lattice.

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We have fermionic creation
and annihilation

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operators
that I'm denoting here with CNC dagger.

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Right. And the number operator n.

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The Hamiltonian has two main terms.

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The first one is a nearest
neighbor hopping.

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So this one will take one electron at site
R prime and hop it onto the site R,

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and we have an interaction term
which takes the form of on site

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repulsion and penalizes
double occupancy of a site.

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There's also a chemical potential term

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which controls the filling of the system.

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This model actually lets us smoothly

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tune between two extremes.

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When T is one,
or T is much larger than u, U is zero

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where in the non-interacting limit,

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as we tune just a little bit of on site
repulsion,

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we enter into a Fermi liquid regime
where perturbation theory applies.

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So all is
well when we take T is equal to zero,

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then it turns out that we're in the flat
band exact flat band regime.

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There's no dispersion.

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And with and interactions fully dominate.

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But and this is the punchline.

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Just having a flat band isn't enough
to guarantee interesting physics,

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as we will see now, both limits,
both the weak, the non-interacting,

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and the exact flat band
limit are exactly solvable,

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and neither captures the richness
that we saw in twisted bilayer graphing.

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So let's begin with a very simple exercise

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and look at the non-interacting
end of the spectrum, right?

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We set the order on site
repulsion to zero.

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00:14:08,966 --> 00:14:11,900
In this case,
the Hubbard model reduces to a textbook

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00:14:11,900 --> 00:14:14,900
tight binding model
that you might remember from undergrad.

240
00:14:15,100 --> 00:14:15,400
Right?

241
00:14:15,400 --> 00:14:18,933
Electrons hop between lattice sites
with a certain amplitude T,

242
00:14:19,566 --> 00:14:21,266
and you can diagonalize this momentum.

243
00:14:21,266 --> 00:14:24,366
And you can diagonalize this Hamiltonian
in momentum space.

244
00:14:25,666 --> 00:14:29,400
The ground state shown here
is just a failed Fermi PSI.

245
00:14:29,433 --> 00:14:29,800
Right.

246
00:14:29,800 --> 00:14:32,000
So you simply take the electrons
and occupy

247
00:14:32,000 --> 00:14:35,533
the lowest available
momentum states up to the Fermi level.

248
00:14:35,866 --> 00:14:38,866
It's completely solvable
and completely benign.

249
00:14:39,000 --> 00:14:43,900
Now if you add interactions,
but they're still weak yet non-zero

250
00:14:44,233 --> 00:14:47,033
when we're in familiar theory,
we can apply many

251
00:14:47,033 --> 00:14:50,600
these perturbation theory
and describe the system in terms of Landau

252
00:14:50,633 --> 00:14:53,633
quasi particles
with renormalized parameters.

253
00:14:54,033 --> 00:14:57,033
This is the Fermi liquid regime
that we talked about earlier.

254
00:14:57,300 --> 00:15:00,533
But now as we move towards
stronger interactions,

255
00:15:00,800 --> 00:15:03,366
this description begins to fail.

256
00:15:03,366 --> 00:15:06,300
This the systems are to starts
to develop correlations

257
00:15:06,300 --> 00:15:08,966
that can't really be captured
by perturbation theory.

258
00:15:08,966 --> 00:15:11,833
So now let's go into the opposite extreme

259
00:15:11,833 --> 00:15:15,100
the flat band limit,
and see what's happening there.

260
00:15:16,666 --> 00:15:17,033
On the

261
00:15:17,033 --> 00:15:20,033
opposite end of the axis
we have the flat band limit.

262
00:15:20,033 --> 00:15:24,200
We set the nearest neighbor
hopping to zero.

263
00:15:24,700 --> 00:15:27,266
In this case,
hopping is completely switched off.

264
00:15:27,266 --> 00:15:30,333
The system breaks up into isolated

265
00:15:30,333 --> 00:15:33,333
lattice sites,
and there's no communication between them.

266
00:15:33,433 --> 00:15:36,333
So the full many body problem

267
00:15:36,333 --> 00:15:40,233
really reduces to solving a single site
Hamiltonian.

268
00:15:41,233 --> 00:15:44,666
Each site has four possible states
I can put.

269
00:15:44,700 --> 00:15:48,700
I can have an empty state,
a spin up state, spin down state, or both.

270
00:15:48,700 --> 00:15:51,233
Electrons, right? Doubly occupied state.

271
00:15:51,233 --> 00:15:56,300
The energy levels are trivial to compute,
and really there's no emergent behavior.

272
00:15:56,300 --> 00:15:59,033
It's just it's just a four level system.

273
00:15:59,033 --> 00:16:01,600
And this is really the key message.

274
00:16:01,600 --> 00:16:04,933
A flat band on its own isn't enough,

275
00:16:05,200 --> 00:16:09,300
even if you have strong interaction,
if the system has no structure.

276
00:16:09,533 --> 00:16:12,500
And I'll explain
exactly what that structure means.

277
00:16:14,100 --> 00:16:16,500
It's basically topology.

278
00:16:16,500 --> 00:16:18,066
There's no connectivity, right?

279
00:16:18,066 --> 00:16:20,266
You don't get anything interesting.

280
00:16:20,266 --> 00:16:24,433
So if you want to explain the richness
of a system

281
00:16:24,433 --> 00:16:27,833
like twisted bilayer graphene,
we need something more.

282
00:16:28,233 --> 00:16:32,266
And the missing ingredient
is actually non-trivial band topology.

283
00:16:33,066 --> 00:16:37,400
And this leads me to the first proper
part of my talk.

284
00:16:37,733 --> 00:16:38,400
Right.

285
00:16:38,400 --> 00:16:41,166
We've seen that neither
strong interactions note

286
00:16:41,166 --> 00:16:44,700
nor flat bands alone are enough to explain

287
00:16:44,700 --> 00:16:47,766
the richness we observe in systems
like twisted bilayer graphene.

288
00:16:48,133 --> 00:16:52,400
So the structure that is missing
is a way to characterize

289
00:16:52,400 --> 00:16:57,500
how bands are built and connected beyond
just their energy, right?

290
00:16:57,933 --> 00:17:01,366
This is exactly where topological quantum
chemistry comes in.

291
00:17:02,466 --> 00:17:06,133
It provides a systematic way to understand

292
00:17:06,133 --> 00:17:10,200
band structures in terms of symmetries
and orbital content,

293
00:17:11,100 --> 00:17:17,800
telling us exactly what a band is made of
and what it's allowed to do right.

294
00:17:17,833 --> 00:17:22,133
In this next section, I'll introduce
some of the key ideas of thick

295
00:17:22,133 --> 00:17:26,400
topological quantum chemistry
and show how they help us navigate

296
00:17:26,733 --> 00:17:31,000
the space of possible flat bands,
especially those that are topological.

297
00:17:32,166 --> 00:17:35,566
Now, depending on your background,

298
00:17:35,933 --> 00:17:40,600
you might look at a material like graphene
from two very different angles.

299
00:17:41,300 --> 00:17:44,866
If you think like a chemist,
you will think about graphene.

300
00:17:44,866 --> 00:17:48,900
In real space you have orbitals,
you have bonds, you have hybridization.

301
00:17:49,200 --> 00:17:53,333
But if you're a physicist,
you are drawn to this spaghetti like plots

302
00:17:53,700 --> 00:17:56,700
which are in reciprocal space, right?

303
00:17:56,800 --> 00:17:59,600
So you introduce concepts
such as Brillouin zones,

304
00:17:59,600 --> 00:18:02,600
dispersions, bands crossing and so on.

305
00:18:03,333 --> 00:18:07,300
But there seems to be
some kind of a conceptual gap, right?

306
00:18:07,600 --> 00:18:10,900
On the left
we have localized atomic orbitals.

307
00:18:11,100 --> 00:18:13,433
Right. Which feel very intuitive.

308
00:18:13,433 --> 00:18:17,666
But on the right we have a band structure
which is completely delocalized.

309
00:18:17,833 --> 00:18:19,866
It's a momentum space object.

310
00:18:19,866 --> 00:18:25,466
Now if we want to understand topology,
this relationship between the real space

311
00:18:25,466 --> 00:18:28,666
and the reciprocal space
is absolutely crucial.

312
00:18:29,400 --> 00:18:32,100
Topology, in a strict sense,

313
00:18:32,100 --> 00:18:36,300
is a global
property of bands in momentum space.

314
00:18:36,300 --> 00:18:40,300
Things like Chern numbers,
which require that you integrate right

315
00:18:40,300 --> 00:18:41,900
across the entire brain. One zone.

316
00:18:43,000 --> 00:18:44,466
But this

317
00:18:44,466 --> 00:18:47,866
but these topological features,
and this is the key idea, really

318
00:18:48,300 --> 00:18:53,433
must have a fingerprint in real space
because electrons really live in orbitals.

319
00:18:53,666 --> 00:18:56,466
So what we need is a formalism
that connects

320
00:18:56,466 --> 00:19:00,533
the real space orbital context to momentum
space topology.

321
00:19:00,533 --> 00:19:03,533
And this is exactly
what topological quantum chemistry does.

322
00:19:04,800 --> 00:19:06,966
So far we've been

323
00:19:06,966 --> 00:19:10,233
I think we've been talking
about band structures.

324
00:19:10,233 --> 00:19:10,600
Right.

325
00:19:10,600 --> 00:19:13,533
In terms of eigenvalues
of certain Hamiltonians.

326
00:19:13,533 --> 00:19:14,600
Right.

327
00:19:14,600 --> 00:19:17,100
But to really get the connection between

328
00:19:17,100 --> 00:19:20,100
real space orbital symmetry and topology,

329
00:19:20,100 --> 00:19:24,200
it would be actually helpful
to model out the energetics, right?

330
00:19:24,466 --> 00:19:27,466
In other words,
let's ask the following question.

331
00:19:27,766 --> 00:19:31,133
Can we talk about band structure

332
00:19:31,133 --> 00:19:34,133
without ever writing down a Hamiltonian?

333
00:19:35,333 --> 00:19:37,733
And the answer is yes, of course.

334
00:19:37,733 --> 00:19:41,666
And the way you do
that is you lean on symmetry

335
00:19:42,000 --> 00:19:46,566
because crystal in materials
are defined by a symmetry group, right?

336
00:19:46,666 --> 00:19:48,300
Something called the space group

337
00:19:48,300 --> 00:19:51,733
which encodes
all the allowed symmetry operation.

338
00:19:51,733 --> 00:19:52,200
Right.

339
00:19:52,200 --> 00:19:55,200
You have translation rotations reflection

340
00:19:55,500 --> 00:19:58,866
and and such and such symmetry operation.

341
00:19:58,866 --> 00:19:59,133
Right.

342
00:19:59,133 --> 00:20:03,466
Given a space group
you need to add only two more ingredients.

343
00:20:03,466 --> 00:20:05,600
Right. To fully characterize the crystal

344
00:20:05,600 --> 00:20:09,533
the position of atoms
within unit cell right.

345
00:20:09,833 --> 00:20:13,533
And the orbitals that each atoms
contribute to the band structure.

346
00:20:13,700 --> 00:20:14,033
Right.

347
00:20:14,033 --> 00:20:18,633
With just these three ingredients
space group, atomic position,

348
00:20:18,633 --> 00:20:23,333
and orbital character, you can ask
what are the possible types

349
00:20:23,333 --> 00:20:26,533
of band structures
that are allowed by symmetry, right?

350
00:20:26,533 --> 00:20:27,533
What are the positive?

351
00:20:27,533 --> 00:20:31,200
What are the possible band structures
that I can get in a real crystal?

352
00:20:31,600 --> 00:20:35,900
And this is the central question
of topological quantum chemistry.

353
00:20:36,600 --> 00:20:39,466
And the answer comes in the form
of something

354
00:20:39,466 --> 00:20:42,466
called elementary band representations.

355
00:20:42,500 --> 00:20:45,166
These are symmetry based building blocks

356
00:20:45,166 --> 00:20:48,166
of all electronic band structures.

357
00:20:48,200 --> 00:20:51,566
So let's let's build
some intuitive picture.

358
00:20:52,033 --> 00:20:54,833
Right. You fixed the space group, right?

359
00:20:54,833 --> 00:20:56,966
Which are the symmetries of a
of a crystal.

360
00:20:56,966 --> 00:21:00,000
But you are still not done
because there are actually there's

361
00:21:00,000 --> 00:21:03,766
a lot of freedom of on
how you place atoms within units.

362
00:21:03,933 --> 00:21:04,733
Right.

363
00:21:04,733 --> 00:21:07,366
And that freedom matters.
Take this example.

364
00:21:07,366 --> 00:21:11,100
All of these three structures here
share the same gravity lattice.

365
00:21:11,366 --> 00:21:13,900
They're built on a triangular lattice.
Right.

366
00:21:13,900 --> 00:21:18,366
But depending on where you put the atoms
you can end up with a triangular lattice,

367
00:21:18,700 --> 00:21:24,033
a honeycomb lattice or a Kagami lattice,
each which each with different numbers

368
00:21:24,033 --> 00:21:29,233
of sides or occupied sides per unit cell
and different symmetry properties.

369
00:21:30,000 --> 00:21:32,366
And the key is that in

370
00:21:32,366 --> 00:21:35,466
all of these cases,
the atoms really are placed

371
00:21:35,466 --> 00:21:38,466
in ways that respect the space group
symmetry of the lattice.

372
00:21:38,733 --> 00:21:41,033
But this specific arrangement, right.

373
00:21:41,033 --> 00:21:44,533
What we call the
the the weak of position, will determine

374
00:21:44,533 --> 00:21:47,533
how the orbitals transform
under the symmetry,

375
00:21:47,866 --> 00:21:50,866
and therefore how the bands
connect in momentum space.

376
00:21:51,233 --> 00:21:54,300
So any theory of elementary band
representations

377
00:21:54,466 --> 00:21:56,366
has to take this into account.

378
00:21:56,366 --> 00:21:59,466
You don't just need the lattice
and the orbitals.

379
00:21:59,466 --> 00:22:01,800
You need to know where these orbitals sit.

380
00:22:03,666 --> 00:22:05,400
Once you fixed the

381
00:22:05,400 --> 00:22:08,400
lattice, once
you fixed the atomic position,

382
00:22:09,266 --> 00:22:11,700
you still have another important detail,
right?

383
00:22:11,700 --> 00:22:17,033
What orbitals you put because different
orbitals transform differently under

384
00:22:17,033 --> 00:22:21,966
the symmetry operation, so they generate
different band structures right on.

385
00:22:22,233 --> 00:22:25,233
In both cases shown here, I've chosen my

386
00:22:25,666 --> 00:22:29,200
occupied lattice
sites to be to form a honeycomb lattice.

387
00:22:29,200 --> 00:22:33,466
Right on the left
we have s orbitals right.

388
00:22:33,633 --> 00:22:36,666
And this should give you something
graphene like right

389
00:22:36,666 --> 00:22:39,666
with Dirac cones and symmetry
and force and force crossings.

390
00:22:40,000 --> 00:22:44,366
However, on the right hand
we play place p and p orbitals instead.

391
00:22:44,633 --> 00:22:47,000
They have very different symmetry
properties, right?

392
00:22:47,000 --> 00:22:49,966
They point in plane,
they will hybridize differently

393
00:22:49,966 --> 00:22:53,233
and the resulting bands
will be behave very differently.

394
00:22:53,400 --> 00:22:56,700
So full theory of band
representation has to consider

395
00:22:56,700 --> 00:23:02,000
not just where the orbitals sit, but also
what kind of orbitals you have right?

396
00:23:03,000 --> 00:23:05,533
The symmetry of the lattice, combined
with the symmetry

397
00:23:05,533 --> 00:23:09,000
of the orbitals, determines
the representation content of the band.

398
00:23:09,833 --> 00:23:14,333
And this is how topological quantum
chemistry builds all of the bands

399
00:23:14,700 --> 00:23:19,666
without Hamiltonian, so not width
of Hamiltonian, but from symmetry.

400
00:23:19,666 --> 00:23:22,233
Ingredients. Right.

401
00:23:22,233 --> 00:23:26,166
So now once I've introduced
all these ingredients,

402
00:23:27,166 --> 00:23:31,866
it's it's time to tell you what the I,
the main idea of topological quantum

403
00:23:31,866 --> 00:23:36,833
chemistry is we want to classify
all possible band structures

404
00:23:36,833 --> 00:23:40,766
that could arise from electron
sitting in a real, in real space orbitals,

405
00:23:41,400 --> 00:23:44,066
right, that obey
crystalline symmetric symmetries.

406
00:23:44,066 --> 00:23:47,600
And we want to do that
without ever invoking a Hamiltonian.

407
00:23:48,400 --> 00:23:50,700
So the way we do that is we

408
00:23:50,700 --> 00:23:54,800
think of bands as being representations
of space group.

409
00:23:55,333 --> 00:23:55,833
Right.

410
00:23:55,833 --> 00:23:59,633
And these representation
are these elementary band representations.

411
00:23:59,966 --> 00:24:03,200
So cut a long story short,
what people have done

412
00:24:03,366 --> 00:24:08,566
and only recently in 2017,
they've taken all the possible space

413
00:24:08,566 --> 00:24:13,900
groups, all the possible ways of arranging
orbitals in them, and they've carefully

414
00:24:13,900 --> 00:24:19,233
tabulated every single band
that can arise in such a system.

415
00:24:19,233 --> 00:24:19,933
Right?

416
00:24:19,933 --> 00:24:25,466
Every single band that can arise
from localized real space orbitals, right.

417
00:24:26,800 --> 00:24:28,866
And then they said,

418
00:24:28,866 --> 00:24:33,366
well, we know all the all the bands
that have

419
00:24:33,366 --> 00:24:38,400
a localized real space description,
and we also know all the possible bands.

420
00:24:38,400 --> 00:24:38,700
Right?

421
00:24:38,700 --> 00:24:41,900
Because we know how bands should connected
momentum space.

422
00:24:42,300 --> 00:24:43,500
Right.

423
00:24:43,500 --> 00:24:45,966
And they reach the following conclusion.

424
00:24:45,966 --> 00:24:50,800
All sets of bands
that are not induced from symmetric

425
00:24:50,800 --> 00:24:55,366
and localized orbitals
are topologically non-trivial by design.

426
00:24:56,066 --> 00:24:59,566
So of course
you can read a lot about topology, right?

427
00:24:59,566 --> 00:25:02,566
You have you have rather complicated

428
00:25:02,800 --> 00:25:05,233
topics such as, Chern numbers.

429
00:25:05,233 --> 00:25:08,233
Right. Invariance in momentum space.

430
00:25:08,266 --> 00:25:11,833
But this gives you
a very simple intuitive explanation.

431
00:25:12,166 --> 00:25:14,866
So let let me visualize that

432
00:25:14,866 --> 00:25:17,866
this diagram shows the the landscape

433
00:25:17,966 --> 00:25:21,100
of all possible band structures
that are allowed by symmetry.

434
00:25:21,566 --> 00:25:24,733
That is, all bands
that satisfy the connection,

435
00:25:24,733 --> 00:25:27,366
the compatibility relations
in momentum space.

436
00:25:27,366 --> 00:25:32,066
Within this full space
we have a distinguished small subset.

437
00:25:32,366 --> 00:25:34,166
These elementary band representation.

438
00:25:34,166 --> 00:25:38,166
And these are band structures
that arise from symmetric

439
00:25:38,166 --> 00:25:42,033
localized orbitals in real space,
in other words, atomic elements.

440
00:25:42,333 --> 00:25:45,900
And here's the punchline not all symmetry

441
00:25:45,900 --> 00:25:49,966
allowed bands
come from a localized real space.

442
00:25:49,966 --> 00:25:50,666
Description.

443
00:25:52,400 --> 00:25:53,466
By definition,

444
00:25:53,466 --> 00:25:56,666
such bands
are topologically non-trivial, right?

445
00:25:56,666 --> 00:26:01,366
And this diagram captures the the logic
of topological quantum chemistry.

446
00:26:01,666 --> 00:26:04,666
Let us have a very simple example.

447
00:26:05,433 --> 00:26:06,000
Right.

448
00:26:06,000 --> 00:26:08,700
Suppose you have a group of bands, right?

449
00:26:08,700 --> 00:26:13,233
And you want to ask are they atomic
or are they topologically right?

450
00:26:13,566 --> 00:26:15,400
The idea of topological quantum

451
00:26:15,400 --> 00:26:19,366
chemistry is to try to express
these bands of interest.

452
00:26:19,366 --> 00:26:19,800
Right.

453
00:26:19,800 --> 00:26:22,800
Which are the field
bands in all these cases

454
00:26:22,833 --> 00:26:26,533
is linear combinations of elementary band
representation.

455
00:26:26,533 --> 00:26:28,133
Right

456
00:26:28,133 --> 00:26:30,366
on the left you have a band structure.

457
00:26:30,366 --> 00:26:34,366
There's just a single elementary band
representation that's atomic.

458
00:26:34,366 --> 00:26:35,866
There's no surprise there, right?

459
00:26:35,866 --> 00:26:40,600
It's induced from localized symmetric
orbitals, and it has an atomic limit.

460
00:26:41,433 --> 00:26:45,000
In the middle,
you have a case where the bands

461
00:26:45,200 --> 00:26:48,200
decompose into a fraction,

462
00:26:48,366 --> 00:26:51,466
like a rational non integer

463
00:26:51,466 --> 00:26:54,466
combination of elementary band
representation.

464
00:26:54,466 --> 00:26:54,966
Right.

465
00:26:54,966 --> 00:26:58,500
By definition, in such a case
your bands are topological.

466
00:26:58,500 --> 00:27:02,933
They do not admit,
a localized real space description.

467
00:27:03,200 --> 00:27:04,100
Right.

468
00:27:04,100 --> 00:27:08,000
And of course you can have
other situation where the bands are

469
00:27:08,000 --> 00:27:11,800
actually an integer linear combination,
but one of the coefficients is negative.

470
00:27:12,000 --> 00:27:16,333
I will not go into the details, but
those are also topologically non-trivial.

471
00:27:17,500 --> 00:27:20,033
So the main takeaway of that

472
00:27:20,033 --> 00:27:24,100
section was that a band is topological

473
00:27:24,333 --> 00:27:29,100
if it doesn't have an X financially
localized real space.

474
00:27:29,100 --> 00:27:31,500
Description. Right.

475
00:27:31,500 --> 00:27:35,266
Armed with this information,
we're finally in a position

476
00:27:35,400 --> 00:27:39,900
to talk about the central question
that I posed at the start of the talk.

477
00:27:40,633 --> 00:27:45,100
Can we realize flat bands, ideally
with non-trivial topology

478
00:27:45,433 --> 00:27:48,433
in conventional crystalline materials?

479
00:27:48,666 --> 00:27:51,066
So until now, we've seen that flat bands

480
00:27:51,066 --> 00:27:54,266
really emerged in either
engineered platforms.

481
00:27:54,266 --> 00:27:54,900
Right.

482
00:27:54,900 --> 00:27:57,633
Your your or with strong magnetic fields.

483
00:27:57,633 --> 00:28:03,033
So we want to replicate the same type
of physics in ordinary honest crystals.

484
00:28:03,033 --> 00:28:03,633
Right.

485
00:28:03,633 --> 00:28:07,300
With materials with no fine tuning,
no mooring,

486
00:28:07,333 --> 00:28:10,333
no external magnetic field, none of that.

487
00:28:11,000 --> 00:28:13,800
Okay, so

488
00:28:13,800 --> 00:28:17,033
why why do we want topological flat bands?

489
00:28:17,033 --> 00:28:17,366
Right.

490
00:28:17,366 --> 00:28:20,366
I gave you this example
with a simple Hubbard model.

491
00:28:20,933 --> 00:28:23,100
But but why do we really

492
00:28:23,100 --> 00:28:26,100
why do we really need
the topological aspects?

493
00:28:26,233 --> 00:28:29,266
This is because not all flat
bands are created equal.

494
00:28:30,100 --> 00:28:32,300
This is where
the topology comes into play.

495
00:28:32,300 --> 00:28:35,966
If your flat band is trivial,
as it happens

496
00:28:35,966 --> 00:28:39,133
here on the on the left hand side right.

497
00:28:39,333 --> 00:28:43,366
If you Fourier transform this flat band,
you get exponentially

498
00:28:43,366 --> 00:28:46,366
localized real space orbitals.

499
00:28:46,400 --> 00:28:50,000
If you try to write down
an interacting theory for that,

500
00:28:50,033 --> 00:28:55,366
for those for that flat band,
what you end up is essentially a term,

501
00:28:55,700 --> 00:29:00,666
a local term that looks like the the
Hubbard interaction term we saw earlier.

502
00:29:01,033 --> 00:29:03,933
So this is so
there is nothing special about this.

503
00:29:03,933 --> 00:29:04,200
Right.

504
00:29:04,200 --> 00:29:06,900
You can still you can decouple the sides

505
00:29:06,900 --> 00:29:09,833
and then you can
you can solve each side separately.

506
00:29:09,833 --> 00:29:11,933
And there's simply nothing special about

507
00:29:13,200 --> 00:29:14,833
about these bands.

508
00:29:14,833 --> 00:29:18,000
In contrast,
if you have a topological flat band

509
00:29:18,866 --> 00:29:23,600
then you really cannot build this
localized real space description either.

510
00:29:23,600 --> 00:29:27,700
You can't have them, either
you can't have the orbitals be symmetric

511
00:29:27,866 --> 00:29:31,733
or in most cases, you can't have them
be exponentially localized.

512
00:29:32,066 --> 00:29:36,100
So when you build an interacting theory
for orbitals

513
00:29:37,466 --> 00:29:42,233
such as these, for a topologically flat
band, you find that the interacting

514
00:29:42,233 --> 00:29:47,000
Hamiltonian is highly non-local,
even though there's no kinetic energy,

515
00:29:47,533 --> 00:29:51,700
the projected interaction is non-local
and the physics is far richer.

516
00:29:52,400 --> 00:29:55,333
And this is exactly why topology matters.

517
00:29:55,333 --> 00:30:00,366
As I showing this schematic
here on on the in the bottom right right.

518
00:30:00,666 --> 00:30:06,033
The most exotic phases in condensed
matter tend to emerge precisely

519
00:30:06,033 --> 00:30:10,733
where this nontrivial band
topology meets strong correlation.

520
00:30:11,100 --> 00:30:13,500
And this is really the target zone.

521
00:30:13,500 --> 00:30:16,566
The upper right
quadrant here of this phase diagram

522
00:30:16,566 --> 00:30:19,566
where both interactions
and topology are strong.

523
00:30:20,066 --> 00:30:24,033
And if you analyze twisted
bilayer graphene, if you analyze the

524
00:30:24,300 --> 00:30:28,666
the flat bands of twisted
bilayer graphene, you will find that

525
00:30:28,666 --> 00:30:32,866
it lives exactly in this in this top right
quadrant.

526
00:30:33,466 --> 00:30:36,200
Right now we're trying to replicate

527
00:30:36,200 --> 00:30:39,400
the same thing in conventional crystals.

528
00:30:39,733 --> 00:30:42,000
How do we how do we even start right.

529
00:30:43,233 --> 00:30:44,166
Let's now look

530
00:30:44,166 --> 00:30:48,000
at, at the concrete example,
the Kagami lattice.

531
00:30:48,000 --> 00:30:51,000
Right. This is this is actually named

532
00:30:51,000 --> 00:30:54,600
after a traditional Japanese basket
weaving pattern.

533
00:30:54,600 --> 00:30:54,900
Right?

534
00:30:54,900 --> 00:30:57,900
I'm showing
here on the on the bottom, left, right.

535
00:30:58,266 --> 00:31:03,000
And at first glance and if you remember
the very beginning of the talk,

536
00:31:03,000 --> 00:31:07,766
you might see you might think, well,
why not just have independent patterns?

537
00:31:07,766 --> 00:31:11,266
Because in the independent
patterns are by definition a flat band.

538
00:31:11,700 --> 00:31:15,633
But then those would be by definition
topologically trivial.

539
00:31:15,766 --> 00:31:18,333
So uninteresting. Right.

540
00:31:18,333 --> 00:31:20,400
So we want to build flat bands

541
00:31:20,400 --> 00:31:23,966
in some sort of a dynamic way
from quantum interference.

542
00:31:24,300 --> 00:31:27,300
Right in the Kagami lattice
that I'm showing here,

543
00:31:28,166 --> 00:31:32,566
you can write something called a compact
localized state.

544
00:31:32,833 --> 00:31:33,733
Right.

545
00:31:33,733 --> 00:31:36,733
And I'm showing this here in red, right.

546
00:31:37,300 --> 00:31:39,966
This is a linear combination of orbitals

547
00:31:39,966 --> 00:31:42,966
with alternating signs around the hexagon.

548
00:31:43,466 --> 00:31:45,300
Now here's the here's the key.

549
00:31:45,300 --> 00:31:49,066
Because of the lattice geometry,
any tunneling

550
00:31:49,266 --> 00:31:53,133
outside of this hexagon exactly
cancels, right?

551
00:31:53,433 --> 00:31:56,100
The destructive interference is perfect.

552
00:31:56,100 --> 00:31:59,666
So this wave function
is an exact eigenstate

553
00:31:59,866 --> 00:32:03,566
that I can write down for every hexagon.

554
00:32:03,800 --> 00:32:08,733
Every hexagonal packet of this kagami,
of this kagami lattice.

555
00:32:08,733 --> 00:32:09,600
Right.

556
00:32:09,600 --> 00:32:12,900
So if you go ahead and, and Fourier

557
00:32:12,900 --> 00:32:17,466
transform and do the proper computation
on the right hand side,

558
00:32:17,500 --> 00:32:22,900
you'll find that you get a flat band right
here, here at energy equal to two.

559
00:32:23,066 --> 00:32:27,300
I'm working in in arbitrary units
where t is equal to one.

560
00:32:28,200 --> 00:32:29,300
Right. But

561
00:32:30,866 --> 00:32:32,333
actually what's what's

562
00:32:32,333 --> 00:32:36,200
even more interesting is if you look
at this band structure on the right,

563
00:32:36,500 --> 00:32:40,366
you see that the flat band
actually touches the dispersive bands

564
00:32:40,733 --> 00:32:43,633
in one single point in the brilliant zone.

565
00:32:43,633 --> 00:32:46,633
This touching
turns out to be protected by symmetry,

566
00:32:46,766 --> 00:32:51,333
and you can't gap it without either
breaking the symmetries of the lattice

567
00:32:51,800 --> 00:32:55,100
or by making the flat band topological.

568
00:32:55,466 --> 00:32:59,800
And this is what I really mean
by this flat bands not admitting

569
00:32:59,800 --> 00:33:03,033
exponentially localized
symmetric quanta orbitals.

570
00:33:03,966 --> 00:33:07,800
At this point,
you might be asking yourself, well, okay,

571
00:33:07,800 --> 00:33:12,433
this is this is all interesting,
but is it really useful for anything?

572
00:33:12,700 --> 00:33:13,766
Right.

573
00:33:13,766 --> 00:33:16,200
And the answer is
of course the affirmative.

574
00:33:16,200 --> 00:33:16,533
Right.

575
00:33:16,533 --> 00:33:21,000
I'm highlighting here
a very recent paper right from last year

576
00:33:21,866 --> 00:33:25,766
where they synthesized
one of these Kagome metals.

577
00:33:26,400 --> 00:33:29,000
So even though this is a 3D

578
00:33:29,000 --> 00:33:32,100
material, it has many active orbitals.

579
00:33:32,600 --> 00:33:36,133
It has chromium, atoms, cesium atoms,

580
00:33:37,266 --> 00:33:39,200
antimony p orbitals.

581
00:33:39,200 --> 00:33:42,666
You can still see here on the right hand

582
00:33:42,666 --> 00:33:45,700
side a clear kagome lattice.

583
00:33:46,000 --> 00:33:50,233
And if you squint at the band structure
here, you can sort of see

584
00:33:50,266 --> 00:33:53,266
near the Fermi level,
you have a flat band,

585
00:33:54,300 --> 00:33:54,900
right?

586
00:33:54,900 --> 00:33:57,900
And in the band structure as a whole

587
00:33:57,900 --> 00:34:01,500
seems to have signatures of the Dirac band
structure.

588
00:34:01,500 --> 00:34:01,966
Right.

589
00:34:01,966 --> 00:34:05,566
So we had
we had this, this Dirac cone here.

590
00:34:05,966 --> 00:34:08,966
We also have it here
we have the flat bands.

591
00:34:09,200 --> 00:34:12,233
Now what's
especially interesting about this

592
00:34:12,233 --> 00:34:15,233
compound is what happens under pressure.

593
00:34:15,266 --> 00:34:15,766
Right.

594
00:34:15,766 --> 00:34:19,266
As shown here on the phase
diagram on the right.

595
00:34:20,166 --> 00:34:20,733
Right.

596
00:34:20,733 --> 00:34:23,800
The system exhibits very robust

597
00:34:24,000 --> 00:34:27,166
superconductivity as I apply,

598
00:34:27,666 --> 00:34:30,433
as I apply pressure directly

599
00:34:30,433 --> 00:34:33,433
above this superconducting phase.

600
00:34:33,533 --> 00:34:36,066
The system is in a non Fermi liquid state.

601
00:34:36,066 --> 00:34:38,733
Now how do I know it's a non Fermi liquid.

602
00:34:38,733 --> 00:34:43,500
Well experiment is can actually measure
the resistance of of a sample.

603
00:34:43,833 --> 00:34:44,266
Right.

604
00:34:44,266 --> 00:34:48,500
And if the resistance
if the system is a Fermi liquid

605
00:34:48,733 --> 00:34:53,066
then then the resistance should various t
squared right where T is the temperature.

606
00:34:53,066 --> 00:34:55,933
Right.
So whenever you see a different exponent.

607
00:34:55,933 --> 00:34:56,733
And this is really

608
00:34:56,733 --> 00:35:00,400
what's being plotted here,
you know that is a non Fermi liquid right.

609
00:35:00,500 --> 00:35:02,033
So this is very interesting.

610
00:35:02,033 --> 00:35:08,466
Very likely superconductivity here
does not come from from phonons.

611
00:35:08,466 --> 00:35:08,833
Right.

612
00:35:10,066 --> 00:35:13,400
It's actually unconventional in nature
and probably akin

613
00:35:13,566 --> 00:35:16,900
akin to one in high temperature
superconductors.

614
00:35:17,600 --> 00:35:19,933
So this is a very beautiful example

615
00:35:19,933 --> 00:35:24,933
of how flat band physics, topology
and correlations all come together

616
00:35:25,266 --> 00:35:28,266
in real materials, not just toy models.

617
00:35:29,100 --> 00:35:33,466
Okay,
but the this Kagami example that I gave

618
00:35:34,200 --> 00:35:37,400
is beautiful, is simple
to understand, simple to explain,

619
00:35:37,666 --> 00:35:42,133
but can actually feel
a little bit mysterious because I seem to

620
00:35:42,700 --> 00:35:46,766
just I seem to have just pulled
this cargo lattice out of my pocket.

621
00:35:46,766 --> 00:35:47,533
Right?

622
00:35:47,533 --> 00:35:51,766
So to really understand
where flat bands can come from,

623
00:35:52,966 --> 00:35:55,700
let me start with a very simple example

624
00:35:55,700 --> 00:35:58,700
and then generalize that to the lattices.

625
00:35:59,266 --> 00:36:02,266
So take a very simple planar molecule.

626
00:36:02,333 --> 00:36:03,266
Right.

627
00:36:03,266 --> 00:36:06,866
Three outer atoms shown in cyan

628
00:36:07,766 --> 00:36:10,966
surrounding a central one shown in orange.

629
00:36:11,466 --> 00:36:11,900
All right.

630
00:36:13,033 --> 00:36:13,766
Now let's

631
00:36:13,766 --> 00:36:16,933
say that I want to build
these are the atomic orbitals

632
00:36:16,933 --> 00:36:20,633
of the molecule right
s orbitals for each atom, right

633
00:36:21,133 --> 00:36:25,500
s orbitals for these triangular atoms,
and one s orbital for the central atoms.

634
00:36:26,066 --> 00:36:28,600
Let's say that I want to build

635
00:36:28,600 --> 00:36:30,800
the molecular orbitals.

636
00:36:30,800 --> 00:36:33,800
Right. How would I do that?

637
00:36:34,033 --> 00:36:36,900
Well I would use concepts
from group theory.

638
00:36:36,900 --> 00:36:37,433
Right.

639
00:36:37,433 --> 00:36:41,100
I can
I can basically take these three atoms,

640
00:36:41,400 --> 00:36:44,566
these three orbitals from the exterior,
make a symmetric,

641
00:36:44,566 --> 00:36:48,166
a c, three symmetric
or a threefold rotation symmetric

642
00:36:48,533 --> 00:36:52,733
combination of them,
and allow those that symmetric combination

643
00:36:52,733 --> 00:36:56,366
to to be combined
with the s orbital on the central atom.

644
00:36:56,700 --> 00:36:57,600
Right.

645
00:36:57,600 --> 00:37:01,266
So these will combine right
and they will split right.

646
00:37:01,266 --> 00:37:03,733
Because this is
this is the avoided crossing row.

647
00:37:03,733 --> 00:37:04,566
Right.

648
00:37:04,566 --> 00:37:09,500
On the other hand these orbitals here
I started with three orbitals.

649
00:37:09,500 --> 00:37:12,500
I must have three after I hybridize them.

650
00:37:12,733 --> 00:37:16,500
These hybridized orbitals here
that I'm denoting with e prime,

651
00:37:16,500 --> 00:37:19,466
which is just notation
from, from group theory.

652
00:37:19,466 --> 00:37:23,900
They really don't have the symmetry
to hybridize with the central.

653
00:37:23,900 --> 00:37:27,900
And so they will just stay here
at zero energy.

654
00:37:28,700 --> 00:37:33,666
Now if you take this very simple picture
put it on a lattice.

655
00:37:34,200 --> 00:37:38,700
It turns out that this gives you
the most general way

656
00:37:39,166 --> 00:37:42,500
of building
flat bands in crystalline materials.

657
00:37:42,733 --> 00:37:46,500
And this is something called
the Li lattice construction.

658
00:37:47,800 --> 00:37:51,933
So in crystal, in flat band materials,

659
00:37:52,433 --> 00:37:57,233
flat bands often appear in systems
called bipartite lattices.

660
00:37:57,666 --> 00:38:01,933
So here's the the idea is it's
actually deceptively simple.

661
00:38:03,300 --> 00:38:05,466
Consider two sub lattices

662
00:38:05,466 --> 00:38:08,466
which I'm going to call L and l tilde.

663
00:38:09,700 --> 00:38:13,500
And they each contain
n l and nll tilde atoms.

664
00:38:13,666 --> 00:38:15,200
Right.

665
00:38:15,200 --> 00:38:17,400
Crucially, I will assume there's

666
00:38:17,400 --> 00:38:21,200
all there isn't any hopping
within the same sub lattice.

667
00:38:21,333 --> 00:38:25,966
So all the hopping the electrons can only
hope from one sub lattice to the other.

668
00:38:26,400 --> 00:38:27,066
Right?

669
00:38:27,066 --> 00:38:32,033
Then a two line
mathematical proof will tell you

670
00:38:32,700 --> 00:38:36,433
that if you have an imbalance
in the number of atoms

671
00:38:36,433 --> 00:38:41,366
in the two sub lattices,
you are guaranteed to get zero

672
00:38:41,366 --> 00:38:45,233
modes, zero modes
which actually become flat bands.

673
00:38:45,233 --> 00:38:47,400
Once you Fourier transform.

674
00:38:47,400 --> 00:38:50,933
And this is exactly what we saw
in this very simple picture, right?

675
00:38:51,100 --> 00:38:56,100
We had three atoms in one of our sub
lattice, one atom in the other.

676
00:38:56,300 --> 00:39:00,766
So we got three minus
2A3 minus one two zero amounts.

677
00:39:00,866 --> 00:39:01,800
Right.

678
00:39:01,800 --> 00:39:04,300
It's the same
game but played on a lattice.

679
00:39:05,766 --> 00:39:08,500
Now, what we did in in

680
00:39:08,500 --> 00:39:11,633
a recent paper
actually was to generalize this,

681
00:39:12,400 --> 00:39:14,800
and, and show

682
00:39:14,800 --> 00:39:18,866
some, some mathematical result
which are very similar

683
00:39:18,866 --> 00:39:22,766
in spirit to,
topological quantum chemistry.

684
00:39:22,900 --> 00:39:27,166
What we essentially showed
that it doesn't matter how you couple

685
00:39:27,166 --> 00:39:31,500
these two sub lattices,
the topological properties of the bands

686
00:39:32,066 --> 00:39:36,033
are only dependent on the atoms
of the two sub lattices.

687
00:39:36,700 --> 00:39:38,000
Right.

688
00:39:38,000 --> 00:39:40,933
And actually,

689
00:39:40,933 --> 00:39:43,933
after our study, a different team

690
00:39:44,600 --> 00:39:48,100
use the same concepts
on all known materials.

691
00:39:48,600 --> 00:39:51,366
Right. And identified 7000.

692
00:39:51,366 --> 00:39:54,333
That's seven times ten to the three

693
00:39:54,333 --> 00:39:58,266
materials, right, that have flat bands

694
00:39:58,766 --> 00:40:01,800
and that you can actually synthesize
in a lab.

695
00:40:02,333 --> 00:40:03,833
Right.

696
00:40:03,833 --> 00:40:08,466
And actually,
you can even show that the Kagami model

697
00:40:08,466 --> 00:40:12,600
I showed previously is actually included
within this formalism.

698
00:40:13,800 --> 00:40:14,466
And, and I'm

699
00:40:14,466 --> 00:40:17,466
showing here in this slide
why that is the case.

700
00:40:17,633 --> 00:40:22,800
If you take the Kagami lattice and call
that your big sub lattice, right.

701
00:40:23,100 --> 00:40:27,200
And, and put in a virtual honeycomb
lattice,

702
00:40:27,566 --> 00:40:31,766
you can see that the same wave function
for the flat band in the Kagami case

703
00:40:32,400 --> 00:40:36,433
is a wave function of the flat band
in the bipartite

704
00:40:36,433 --> 00:40:40,033
crystalline lattice case,
and the reasoning is very similar, right?

705
00:40:40,366 --> 00:40:46,300
But now with this sort of bipartite
formulation, I can get all these results

706
00:40:46,466 --> 00:40:50,700
without ever invoking any sort of hopping
hopping whatsoever.

707
00:40:51,900 --> 00:40:54,133
In this example here,

708
00:40:54,133 --> 00:40:58,266
I'm actually building
two toy models of flat bands.

709
00:40:58,466 --> 00:40:59,266
Right?

710
00:40:59,266 --> 00:41:02,633
So in both cases I take my big sub lattice

711
00:41:02,633 --> 00:41:05,700
to be made up of s orbitals
on a kagami lattice.

712
00:41:05,700 --> 00:41:07,400
Right. These black dots.

713
00:41:07,400 --> 00:41:10,400
And then I vary the

714
00:41:10,433 --> 00:41:13,433
the contents of the small subplots.

715
00:41:13,533 --> 00:41:15,733
To cut a long story short,

716
00:41:15,733 --> 00:41:18,733
on the left hand side I can go ahead.

717
00:41:19,000 --> 00:41:21,933
Use topological quantum chemistry
once again,

718
00:41:21,933 --> 00:41:24,933
not do a single calculation

719
00:41:25,366 --> 00:41:28,800
and show that the flat band
must be topologically non-trivial

720
00:41:29,666 --> 00:41:31,866
and on the right hand side,

721
00:41:31,866 --> 00:41:34,866
using the same algorithmic prescription,

722
00:41:34,866 --> 00:41:37,400
I can do the same and show
that the flat band

723
00:41:37,400 --> 00:41:41,966
has a symmetry protected gapless point
at the gamma point, right?

724
00:41:41,966 --> 00:41:45,566
So it touches the dispersive
band at the gamma point.

725
00:41:46,733 --> 00:41:50,100
And this once again, this is not just,

726
00:41:50,100 --> 00:41:54,300
this is not just just a mathematical
toy model.

727
00:41:54,600 --> 00:41:58,033
You can actually go ahead
and apply this to real crystals.

728
00:41:58,433 --> 00:42:01,100
And, and we we've actually done that.

729
00:42:01,100 --> 00:42:04,100
I'm highlighting here one of the compounds

730
00:42:04,200 --> 00:42:07,200
that shows a flat band
with a band touch point

731
00:42:07,466 --> 00:42:11,266
that can be diagnosed diagnose with,
with this type of prescription.

732
00:42:11,666 --> 00:42:16,000
Actually if you want,
you can go ahead and scan this QR code

733
00:42:16,700 --> 00:42:21,100
and it points to an online database
of flat bands.

734
00:42:21,666 --> 00:42:25,266
And in your spare time, you might want to
synthesize one of those compounds.

735
00:42:25,633 --> 00:42:30,000
My personal recommendation is once
you go to the website,

736
00:42:30,233 --> 00:42:33,566
you should go to this best flat
bands, section.

737
00:42:33,933 --> 00:42:38,366
Because these have the cleanest, 
the cleanest flat bands

738
00:42:38,700 --> 00:42:41,433
and actually one of the one

739
00:42:41,433 --> 00:42:45,866
another recent experimental
paper, this time from MIT,

740
00:42:46,800 --> 00:42:47,766
they took one of the

741
00:42:47,766 --> 00:42:51,200
compounds
that was highlighted in this database.

742
00:42:51,833 --> 00:42:55,066
The compound looks something that,

743
00:42:55,066 --> 00:42:59,700
you would never write in a chemistry
exam is calcium nickel two.

744
00:43:00,400 --> 00:43:01,166
Right.

745
00:43:01,166 --> 00:43:05,066
And the interesting thing about this,
this compound is that it

746
00:43:05,066 --> 00:43:08,100
actually has flat
bands close to the Fermi level.

747
00:43:08,533 --> 00:43:09,133
Right?

748
00:43:09,133 --> 00:43:15,433
So there are here on on in panel
A, I'm showing the result of our pitch.

749
00:43:15,500 --> 00:43:20,233
That's that's angle
resolved, for the emission spectroscopy.

750
00:43:21,000 --> 00:43:21,600
Right.

751
00:43:21,600 --> 00:43:27,133
And this essentially probes
the band structure of an actual crystal.

752
00:43:27,500 --> 00:43:28,033
Right.

753
00:43:28,033 --> 00:43:32,433
And you can see here that
they've identified this flattish bands.

754
00:43:32,966 --> 00:43:33,433
Right.

755
00:43:33,433 --> 00:43:35,900
And they they even compare them to theory.

756
00:43:35,900 --> 00:43:38,600
Now, if you have these flat bands quite

757
00:43:38,600 --> 00:43:42,300
far from the Fermi energy,
then nothing really happens.

758
00:43:42,466 --> 00:43:45,466
But if you take the compound, you dope it.

759
00:43:45,766 --> 00:43:49,800
Doping means that you add different atoms
with different number of electrons.

760
00:43:51,333 --> 00:43:52,933
You see that exactly

761
00:43:52,933 --> 00:43:55,933
when the flat band touches
the Fermi energy,

762
00:43:56,766 --> 00:43:59,533
you actually get superconductivity

763
00:43:59,533 --> 00:44:03,400
developing, just like we did in twisted
bilayer graphene.

764
00:44:03,666 --> 00:44:08,666
Now, I admit that six Kelvin,
which is the critical temperature

765
00:44:08,666 --> 00:44:12,733
of this superconductor,
is still not not enough.

766
00:44:12,933 --> 00:44:16,900
But considering the time scale
right on which on which

767
00:44:16,900 --> 00:44:21,033
these experiments happen, I would say that
it's it's quite impressive.

768
00:44:21,633 --> 00:44:24,633
And before finishing,

769
00:44:24,666 --> 00:44:29,100
I just want to say that, maybe

770
00:44:29,466 --> 00:44:34,033
discarding
topologically trivial bands is not,

771
00:44:35,400 --> 00:44:38,133
they should not be discarded
straight away.

772
00:44:38,133 --> 00:44:41,133
In fact, actually, in a recent work,

773
00:44:41,566 --> 00:44:45,833
we've shown that there's,
new, fundamentally different,

774
00:44:46,200 --> 00:44:51,266
universality class of moiré materials

775
00:44:52,633 --> 00:44:56,666
in which the moral potential acts
essentially as a magnetic field

776
00:44:57,366 --> 00:45:00,366
and realize is a very interesting algebra

777
00:45:00,433 --> 00:45:03,366
of the mirror and translation operators.

778
00:45:03,366 --> 00:45:05,866
Essentially,
after you go through all these maths,

779
00:45:06,900 --> 00:45:08,233
you are able to show

780
00:45:08,233 --> 00:45:13,300
that this flat bends in twisted
thin selenite, right?

781
00:45:13,300 --> 00:45:18,666
They have a very interesting quasi
one dimensional property.

782
00:45:18,866 --> 00:45:21,866
So even though the system is is two
dimensional,

783
00:45:22,200 --> 00:45:25,533
it's actually made up of one dimensional
chains.

784
00:45:25,700 --> 00:45:29,000
Electrons
cannot hop between chains, right?

785
00:45:29,000 --> 00:45:32,866
But they still interact in electrostatics
between chains.

786
00:45:33,166 --> 00:45:36,200
So this
this actually paves the road to the,

787
00:45:36,200 --> 00:45:39,200
for example, realizing
something called a Latin jr liquid.

788
00:45:39,266 --> 00:45:39,566
Right.

789
00:45:39,566 --> 00:45:42,600
And and really studying
like low dimensional physics

790
00:45:42,833 --> 00:45:45,300
in, in real experiments.

791
00:45:45,300 --> 00:45:47,733
So with that I'm going to leave you

792
00:45:47,733 --> 00:45:51,000
with conclusions
and only point one more thing.

793
00:45:51,733 --> 00:45:55,200
The fact that
because we know flat band compound

794
00:45:55,300 --> 00:45:59,533
compounds
don't obey Fermi liquid theory, right?

795
00:45:59,533 --> 00:46:03,600
We need to come up with,
with new tools on how to solve them.

796
00:46:04,200 --> 00:46:08,833
And one of one of one possible
such new tool

797
00:46:09,400 --> 00:46:12,666
is actually going to be highlighted
by the next talk.

798
00:46:12,666 --> 00:46:15,666
So make sure you stick for
Dominique Stock.

799
00:46:15,966 --> 00:46:16,366
Thank you.

800
00:46:27,733 --> 00:46:29,333
So you're asking up.

801
00:46:29,333 --> 00:46:30,333
Yeah. Yeah.

802
00:46:30,333 --> 00:46:35,366
You're asking outside the perturbative
regime if there are any dualities.

803
00:46:35,633 --> 00:46:36,600
Dualities between.

804
00:46:36,600 --> 00:46:37,233
Between what?

805
00:46:37,233 --> 00:46:40,233
Between something like this.

806
00:46:41,700 --> 00:46:44,233
Well,
I know that people are looking into that.

807
00:46:44,233 --> 00:46:47,400
I don't know specifically how relevant

808
00:46:47,400 --> 00:46:50,700
those are for real materials,
you know, but it's it's

809
00:46:50,700 --> 00:46:54,633
definitely a very interesting,
interesting avenue to to pursue.

810
00:46:54,833 --> 00:46:55,100
Yeah.

811
00:46:55,100 --> 00:46:56,466
But off the top of my head,

812
00:46:56,466 --> 00:47:00,233
I don't know of any such duality relevant
for real materials.

813
00:47:00,400 --> 00:47:03,233
It's a it's a constantly evolving field.

814
00:47:03,233 --> 00:47:03,600
Right.

815
00:47:03,600 --> 00:47:06,966
So what isn't around today
could very well be around tomorrow.

816
00:47:08,033 --> 00:47:08,466
Yeah.

817
00:47:08,466 --> 00:47:11,366
So so the question is really about

818
00:47:11,366 --> 00:47:16,266
how about the experimental difficulties
of actually realizing these platforms.

819
00:47:16,700 --> 00:47:21,766
Well, I must say that, well,
there are problems

820
00:47:21,766 --> 00:47:25,966
with, with growing crystalline materials
that experimentalists have,

821
00:47:26,533 --> 00:47:30,633
creatively, managed to tackle.

822
00:47:31,000 --> 00:47:34,333
There's still a problem
of actually doping the materials,

823
00:47:34,566 --> 00:47:38,100
because maybe you get a flat band
material, but the flat band is far

824
00:47:38,100 --> 00:47:39,333
from the Fermi energy.

825
00:47:39,333 --> 00:47:43,566
You need to dope it,
and people just have to be creative.

826
00:47:43,566 --> 00:47:46,833
They have to try a lot of things,
because sometimes you just add the dopant

827
00:47:47,133 --> 00:47:50,800
and the dopant doesn't diffuse
into the system, right?

828
00:47:50,800 --> 00:47:54,866
It just stays in one place and it's
actually an impurity rather than a dopant.

829
00:47:55,100 --> 00:47:55,433
Right.

830
00:47:55,433 --> 00:47:58,600
So it's it's it's
not really it's not really.

831
00:47:58,733 --> 00:48:02,100
But there aren't 3D recipes for that.

832
00:48:02,333 --> 00:48:05,333
And this is why, compared to,

833
00:48:05,400 --> 00:48:08,400
to engineered hetero structures.

834
00:48:08,433 --> 00:48:12,600
Well,
you can literally peel sheets of graphene,

835
00:48:12,600 --> 00:48:15,633
stack them on top of one another,
put them in a dual

836
00:48:15,633 --> 00:48:18,866
gated setup, and just play
with the electric field to dope them.

837
00:48:19,266 --> 00:48:23,866
For these type of conventional crystals,
there is a lot more work involved.

838
00:48:24,233 --> 00:48:27,700
And that's
why that's why results are probably,

839
00:48:28,100 --> 00:48:32,300
experimental results are
probably appearing at a much slower pace.

840
00:48:32,466 --> 00:48:36,466
But people are working and people
are getting more and more creative.

841
00:48:36,733 --> 00:48:39,733
And, now probably

842
00:48:39,833 --> 00:48:42,833
sort of catalyzed by twisted
bilayer graphene,

843
00:48:42,966 --> 00:48:47,900
they are understand
that this is something worthwhile.

844
00:48:48,200 --> 00:48:51,200
And so
so they're dedicating more time to it.

845
00:48:51,266 --> 00:48:52,533
And yes.

846
00:48:52,533 --> 00:48:55,533
So okay, so the the question is,

847
00:48:55,533 --> 00:48:58,533
where does
where do the energetics come into play?

848
00:48:58,900 --> 00:49:02,100
And for that, people have,

849
00:49:02,333 --> 00:49:05,666
have developed computational tools

850
00:49:06,000 --> 00:49:10,166
which for the vast
majority of materials are remarkably good.

851
00:49:10,166 --> 00:49:14,200
And I'm speaking, of course, about, 
density functional theory.

852
00:49:14,800 --> 00:49:15,133
Right.

853
00:49:15,133 --> 00:49:19,533
So it's these I'm not the DFT, the person.

854
00:49:19,533 --> 00:49:22,533
But the way I understand it, I understand
it is that,

855
00:49:22,833 --> 00:49:27,333
the programs are so powerful,
you're just feeding some atom position

856
00:49:27,333 --> 00:49:30,333
and telling them what type of atoms
you have and the distances.

857
00:49:30,333 --> 00:49:33,333
Then they they produce a band structure.

858
00:49:33,600 --> 00:49:37,700
Oftentimes, it can happen
that the band structure is not accurate,

859
00:49:37,700 --> 00:49:40,100
especially if you have
strong correlations.

860
00:49:40,100 --> 00:49:44,700
So then you actually need to
you actually need to to get creative

861
00:49:44,700 --> 00:49:48,266
right with, with post-processing this DFT
calculations.

862
00:49:48,266 --> 00:49:48,900
Right.

863
00:49:48,900 --> 00:49:52,066
Sometimes sometimes the, the way you do

864
00:49:52,066 --> 00:49:55,766
that is while you rely
on some experimental input.

865
00:49:55,766 --> 00:49:56,200
Right.

866
00:49:56,200 --> 00:50:00,200
And you, you try to work out
what kind of approximation

867
00:50:00,500 --> 00:50:06,233
can make your, energetics agree
with, with experimental measurements.

868
00:50:06,433 --> 00:50:06,733
Right.

869
00:50:06,733 --> 00:50:11,966
But there are tools for, for, 
finding out what the energetics are.

870
00:50:12,233 --> 00:50:15,366
The problem is that, you know,

871
00:50:15,366 --> 00:50:19,266
oftentimes this
these tools are a bit like black boxes.

872
00:50:19,266 --> 00:50:22,266
You can't just
you can't just randomly run,

873
00:50:23,000 --> 00:50:26,233
right, some, some atomic arrangement
and hope that it works.

874
00:50:26,233 --> 00:50:29,600
And this is where the topology insight
comes into play, right.

875
00:50:29,600 --> 00:50:33,533
Because it tells you, okay,
out of the myriad of, out of the infinite

876
00:50:33,533 --> 00:50:37,066
possibilities you could do, here's
what is really worthwhile.

877
00:50:37,600 --> 00:50:38,666
Yeah, yeah.

878
00:50:38,666 --> 00:50:41,100
So the question
the question is, have people

879
00:50:41,100 --> 00:50:46,100
if people thought about the mixed,
the mixed momentum space,

880
00:50:46,100 --> 00:50:49,100
real space, the present day
or formulation of quantum mechanics,

881
00:50:49,100 --> 00:50:52,800
and try to apply that in the context
of topological band theory.

882
00:50:53,566 --> 00:50:58,133
So the very honest answer to
that is not, to my knowledge,

883
00:50:58,400 --> 00:51:02,100
and quite frankly,
the only interaction I had with

884
00:51:02,100 --> 00:51:05,966
this mixed representation
was during my third year of undergrad.

885
00:51:06,133 --> 00:51:09,800
So there's quite a there's quite some time
from, from then, but,

886
00:51:10,266 --> 00:51:14,300
it might be something worthwhile,
especially for non-interacting

887
00:51:14,300 --> 00:51:15,833
band theory.

888
00:51:15,833 --> 00:51:19,400
It might, it might actually
it might turn out to be insightful.

889
00:51:19,700 --> 00:51:22,466
So maybe, maybe people will look at that.

890
00:51:22,466 --> 00:51:22,766
Yeah.

891
00:51:22,766 --> 00:51:25,766
But to my knowledge, I'm
not aware of anything.

892
00:51:25,800 --> 00:51:27,633
Well, actually. Okay.

893
00:51:27,633 --> 00:51:30,633
So that's
that's a very interesting observation.

894
00:51:30,800 --> 00:51:35,066
And it's, it's related to something
that we've done recently.

895
00:51:35,233 --> 00:51:38,533
It turns out that some of these
topological flat bands.

896
00:51:38,533 --> 00:51:38,866
Right.

897
00:51:38,866 --> 00:51:43,266
You can't really
you can't really, trivialize them.

898
00:51:43,566 --> 00:51:46,466
But you can do a basis transformation

899
00:51:46,466 --> 00:51:49,466
and actually convert them
into a heavy fermion model.

900
00:51:50,166 --> 00:51:53,800
So there is a duality
between some of these flat bands and,

901
00:51:53,800 --> 00:51:57,766
and heavy fermions, as a matter of fact,
twisted bilayer graphene.

902
00:51:57,766 --> 00:52:01,766
You can do, an honest
to God unitary transformation

903
00:52:01,766 --> 00:52:04,900
on the band structure and map that
to a topological heavy fermion.

904
00:52:05,066 --> 00:52:09,033
So then you can actually use a lot
of the techniques techniques from there.

905
00:52:09,233 --> 00:52:13,966
Now, in for this specific experiment.

906
00:52:14,166 --> 00:52:14,566
Right.

907
00:52:14,566 --> 00:52:18,300
Obviously there are no electrons in this,
in this compound.

908
00:52:18,300 --> 00:52:23,466
So it's not a heavy fermion, but it could
very well be that you can take

909
00:52:23,466 --> 00:52:27,866
some of these, you can you can take
some of the momentum states, right.

910
00:52:27,900 --> 00:52:32,800
Do some band inversions
and, and construct f like, electron

911
00:52:32,800 --> 00:52:34,466
state for the flat band

912
00:52:34,466 --> 00:52:38,700
and then re express the same sort of
physics in the language of heavy fermions.

913
00:52:38,700 --> 00:52:40,900
But that's definitely something possible.

914
00:52:40,900 --> 00:52:43,900
And people have have looked into that
into great detail.

915
00:52:44,233 --> 00:52:45,366
And thank you.

916
00:52:46,800 --> 00:52:47,200
Okay.

917
00:52:47,200 --> 00:52:49,766
I think we have to
move on to that specter, which we again.