The Wilson loop method provides a powerful tool to understand topological properties of materials by analyzing the evolution of the "charge center" of wave functions across the Brillouin zone, directly revealing bulk topology and predicting the behavior of boundary states.
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[Music]
now we are going to talk about another
method to identify the topology that is
the wilson loop method
with this method we will have a deeper
understanding about the spark boundary
correspondence let's start again from a
two dimensional insulator
and this insulator
here we plot the band structure in this
k x ky plane
we know if we cut a line along k y here
it's also an insulator but it is
one-dimensional if we integrate the
block wheel function along this line
for this quantity actually this is the
bearish surface it corresponding to the
linear charge center of the valence band
here because theta is the linear charge
center is corresponding to the inner
coordinate of the valence band wave function
function and
and
here you see it's also a wave function
of k x because at a different k x this
insulator cut this line are different
now let's plot theta as a function of kx
and we will see that when your charge
center at different k x are slightly
different it has evolution
because it's periodic it repeats
along the y
between zero and one and two higher to
two three
it is also periodic along kx
and therefore if we start
from minus point five the boundary of
the berlin zone to another boundary it
will recover itself therefore
therefore
from the left end to the right end
this line this evolution should should
be a smooth evolution
but there's only two ways to connect
them for a two band model one way is
this zero connect to another zero smoothly
smoothly
the second possibility is actually is
from this part of the periodicity
connect to this one or opposite or from
this zero to another one
clearly we see there's a difference of
topology and if we
science this is periodic for example
along the y axis if we draw this
periodic boundary condition into a
cylinder we see it better you see here
along x this cylinder
then the first one is topologically
different from the second one because
the second one has a winding around the cylinder
cylinder
clearly see they are topologically
different but how this topology how is
connected to the boundary yeah we always
talk about this bulk of boundary correspondence
correspondence
let's look at it suppose
this is the veneer center evolution and
it's exactly the same one for the right
one we call it chain number one later we
will understand why it's chain number one
one
uh here in this case
the evolution is here every yellow box
is the periodicity along y direction and
if we look at the real space suppose we
have an electron travel along kx travel
along x from right to the left i think
the consequences is also shift along the
y axis right it will shift like one
shift to another unit c or the second
unit cell to higher higher higher than
the consequences you see immediately if
we have a boundary this electron has
nowhere to go here we have accumulated
two electrons but at this boundary
there's no electron left
here we call it the charge pumping
and the charge pumping is directly the
consequence of this linear charge center evolution
evolution
and it also tells us on the boundary we
will have about some special boundary
states different from bulk and
and yeah
yeah
this evolution
usually also called this wilson loop
evolution if we have many bands this
evolution can be more complicated but
the con
is general feature is the boundary will
become very different from the bulk
and one more feature you will see is
actually the pattern of this evolution
will be exactly the surface state the
boundary state dispersion now let's go
back to the same band structure and
this one the evolution this is the true
case that one corresponding to that we
have a winding case this binding case
you see
if we go to calculate really a boundary
state the edge state that is here is
exactly that one this is the edge
dispersion this is the insulator the
bulk states but inside the bulk gap we
have a chiral edge this chiral edge
means here the electron only moves to
one direction and
and
more important what you see is the
wilson loop shape is
is
topologically identical to the edge
state dispersion
and this is the case for a chain
insulator but if we have a topological
insulator like this z2 number then
then
actually why it's called z2 number by
because here we have only two
possibilities for the topological insulator
insulator
and we said this mu zero either zero or
one right or in the wilson loop language
here it tells us if we have time versus symmetry
symmetry
this timers invariant point kx0 or the
zoom boundary we have time loss of
symmetry and they always have this
crevice degeneracy this double
degeneracy because the valence band is
double degenerate it's one year charge
center also double degenerate
this is the basic requirement like
from this point to the middle there are
two double degenerate one is centered
now let's connect them a trivial we
connected them is these two start they
meet again in the middle
and there's another possibilities these
two this plate they meet with another
partner in this sense between two timers
or invariant point this crammers partner
they just switch to each other and this
switch give us a topological change
and you see the edge states actually the
edge stays has exactly the same shape as
this veneer center and
and
please remember the edge stays is a
direct calculation of the edge structure
here this topological environment this
this wilson loop just a calculation from
the bulk band structure is from the
spark not that edge
in this sense we understand the boundary
we know how the bulk ever
evolved we know the bulk topology we
will know there will be a special
surface states we even know how this the
surface state dispersion looks like
let's go to a more realistic material
actually this is a band structure or
business selenide this is a bulk band structure
structure
if we
massage this band structure or
re-understand this band structure
from the
viewpoint of the waste and loop the one
near charge center this is what we get
the waste loop for all the well spent
because we have many well expand
therefore we have many
you see many dispersions in this version loop
loop
this version loop evolution you see
exactly you see the dirac coil you see
the clear direction on the surface there
are dirac one too and
and
these are very beautiful this is a very
beautiful example of this bulk boundary correspondence
correspondence
now we know this wilson loop method
simply from the bulk band structure we
get this wilson loop or in other words
with loop include the message of the
wave function give us a different angle
to look at the band structure we know
immediately the topology not only the
bug topology also we know immediately
how the surface state disperses
and so far everything is prepared
we know the dft method how to calculate
the band structure we know the bending
version parity and waste loop then how
to identify the topology in the next
lectures we will go to the realistic
materials for topological insulator
topological scent metals
and then there we investigate the real material
material [Music]
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