Download Integer Quantum Hall Effect - (Dawn of topology in

Integer Quantum Hall Effect
(Dawn of topology in electronic structure)
Raffaele Resta
Dipartimento di Fisica, Università di Trieste
Trieste, May 2016
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Outline
1
What topology is about
2
Topology shows up in electronic structure
3
Classical Hall effect
4
2d noninteracting electrons in a magnetic field
5
Quantum Hall Effect
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Topology
Branch of mathematics that describes properties which
remain unchanged under smooth deformations
Such properties are often labeled by integer numbers:
topological invariants
Founding concepts: continuity and connectivity, open &
closed sets, neighborhood......
Differentiability or even a metric not needed
(although most welcome!)
.
.
.
.
.
.
Topology
Branch of mathematics that describes properties which
remain unchanged under smooth deformations
Such properties are often labeled by integer numbers:
topological invariants
Founding concepts: continuity and connectivity, open &
closed sets, neighborhood......
Differentiability or even a metric not needed
(although most welcome!)
.
.
.
.
.
.
thors Pictures
onardo Da Vinci
Shop
mposers Posters
mposers T shirts
mous Physicists
Math Mug (set theory):
If you consider the set of all sets that have
never been considered ....
A coffee cup and a doughnut are the same
The above shapes are topologically equivalent
and are of Genus 0
Math Mug - Topology
To a Topologist This is a Doughnut
http://cosmology.uwinnipeg.ca/Cosmology/Properties-of-Space.htm (7 of 11) [03/01/2002 7:23:17 PM]
Topological invariant: genus (=1 here)
Math Mug:
Real Life is a Special Case
(black background)
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Gaussian curvature: sphere
In a local set of coordinates in the
tangent plane
z =R−
(
Hessian
H=
Gaussian curvature
√
R2 − x 2 − y 2 ≃
1/R
0
0
1/R
)
K = det H =
.
x2 + y2
2R
.
1
R2
.
.
.
.
Gaussian curvature: sphere
In a local set of coordinates in the
tangent plane
z =R−
(
Hessian
H=
Gaussian curvature
√
R2 − x 2 − y 2 ≃
1/R
0
0
1/R
)
K = det H =
.
x2 + y2
2R
.
1
R2
.
.
.
.
Positive and negative curvature
Bob Gardner's "Relativity and Black Holes" Special Relativity
>
For example, a plane tangent to a 2/3/12
sphere
lies
the
sphere,
and and
so Black
a sphere
is of Relativity
positive
Bob
Gardner's
"Relativity
Holes" Special
4:27
PM entirely on one side of
curvature. In fact, a sphere of radius r is of curvature 1/r2.
2/3/12 4:27 PM
>
A cylinder
is of zero
curvature
since a tangent plane lies on one side of the cylinder and the points of
saddle
shaped
(or more
precisely, a hyperbolic paraboloid)
is of negative
curvature.
A tangent
For example, a plane tangent to a sphere lies entirely on one side ofAthe
sphere,
and surface
so a sphere
is of positive
inof
blue)
are a lineare
containing the point of tangency. Also, of course, a plane is of zero
plane lies on both sides of the surface. Here, the point of tangency isintersection
red and the(here
points
intersection
curvature. In fact, a sphere of radius r is of curvature 1/r2.
curvature.paraboloid.
blue. Pringles potato chips are familiar examples of sections of a hyperbolic
Smooth surface, local set of coordinates on the tangent plane
)
( 2
2
K = det
http://www.etsu.edu/physics/plntrm/relat/curv.htm
A saddle shaped surface (or more precisely, a hyperbolic paraboloid) is of negative curvature. A tangent
plane lies on both sides of the surface. Here, the point of tangency is red and the points of intersection are
blue. Pringles potato chips are familiar examples of sections of a hyperbolic paraboloid.
∂ z
∂x 2
∂2z
∂y ∂x
∂ z
∂x∂y
∂2z
∂y 2
Page 6 of 9
If two surfaces have the same curvature, we can smoothly transform one into the other without changing
distances (the transformation is called an isometry). For example, a sheet of paper (used here to represent a
curvature zero plane) can be rolled up to form a cylinder (which also has zero curvature). However, we
cannot role the paper smoothly into a sphere (which is of positive curvature). For example, if we try to
giftwrap a basketball, then the paper will overlap itself and have to be crumpled. We also cannot role the
.
.
.
.
.
.
paper smoothly over a saddle shaped surface (which is of negative curvature) since this would require ripping
a curved space but not in a flat space may lead to the idea of using it as a way to
measure curvature. We cannot prove it mathematically here, but it turns out that if we
choose a loop that is small enough around a point in a curved space, the amount of
change in the direction of a vector that is parallel transported along it is proportional
to the area enclosed by the loop. So, the ratio between the area of the loop and the
amount of change in the direction of the vector (whatever way we chose to measure
it) can be used as a measure to the curvature of the surface that includes the loop.
Actually we define curvature by the value of this ratio.
Twist angle vs. Gaussian curvature
Gaussian curvature of the
spherical surface Ω = 1/R 2
Angular mismatch for parallel transport:
∫
Riemann Curvature Tensor
γ = dσ K
Riemann tensor is a rank (1,3) tensor that describes the curvature in all directions at a
given point in space. It takes 3 vectors and returns a single vector. The vectors that are
fed to the tensor should be very small and have a length ε. If we use the first two
vectors to form a tiny parallelogram and we parallel transport the third vector around
this parallelogram, the vector that the function returns is approximately the vector
difference between the original vector and the vector after the parallel transport.
Mathematically it can be written like this:
Equivalently: sum of the three angles:
∫
α1 + α2 + α3 = π + γ = π +
.
dσ K
.
.
.
.
.
a curved space but not in a flat space may lead to the idea of using it as a way to
measure curvature. We cannot prove it mathematically here, but it turns out that if we
choose a loop that is small enough around a point in a curved space, the amount of
change in the direction of a vector that is parallel transported along it is proportional
to the area enclosed by the loop. So, the ratio between the area of the loop and the
amount of change in the direction of the vector (whatever way we chose to measure
it) can be used as a measure to the curvature of the surface that includes the loop.
Actually we define curvature by the value of this ratio.
Twist angle vs. Gaussian curvature
Gaussian curvature of the
spherical surface Ω = 1/R 2
Angular mismatch for parallel transport:
∫
Riemann Curvature Tensor
γ = dσ K
Riemann tensor is a rank (1,3) tensor that describes the curvature in all directions at a
given point in space. It takes 3 vectors and returns a single vector. The vectors that are
fed to the tensor should be very small and have a length ε. If we use the first two
vectors to form a tiny parallelogram and we parallel transport the third vector around
this parallelogram, the vector that the function returns is approximately the vector
difference between the original vector and the vector after the parallel transport.
Mathematically it can be written like this:
Equivalently: sum of the three angles:
∫
α1 + α2 + α3 = π + γ = π +
.
dσ K
.
.
.
.
.
choose a loop that is small enough around a point in a curved space, the amount of
change in the direction of a vector that is parallel transported along it is proportional
to the area enclosed by the loop. So, the ratio between the area of the loop and the
amount of change in the direction of the vector (whatever way we chose to measure
it) can be used as a measure to the curvature of the surface that includes the loop.
Actually we define curvature by the value of this ratio.
What about integrating over a closed surface?
Figure 4. The Gauss–Bonnet formula
trated here by a toroidal surface with
local curvature K is positive on those
face that resemble a sphere and nega
the hole, that resemble a saddle. Bec
of handles, equals one, the integral o
the entire surface vanishes. One can
Shiing-shen Chern’s quantum genera
Gauss–Bonnet formula plausible by c
angular mismatch of parallel transpo
around the small red patch in the fig
allel transport around the small loop enclosing the area
dA. The left side of the Gauss–Bonnet equation is geometric and not quantized a priori. But the right side is
manifestly quantized; the integer g is the number of handles characterizing the topology of S. (For the torus in figure 4, g = 1.) So, if we change K arbitrarily by denting the
Riemann Curvature Tensor
surface—so long as we do not punch through new handles—the quantized right side of equation 3 does not
Riemann tensor is a rank (1,3) tensor that describes the curvature in
all directions at a
change.
The
Gauss–Bonnet
given point in space. It takes 3 vectors and returns a single vector. The
vectors
that areformula has a well-known modern
generalization, due to Shiing-shen Chern. The Gaussfed to the tensor should be very small and have a length ε. If we
use the first
twoapplies not only to the geometry of
Bonnet-Chern
formula
surfaces
but
also
to the geometry of the eigenstates paravectors to form a tiny parallelogram and we parallel transport the meterized
third vector
around
by F and q. It looks exactly like equation 3, exthis parallelogram, the vector that the function returns is approximately
vector
cept that Kthe
is now
the adiabatic curvature of equation 1,
the surface
S is a torus parameterized by the two
difference between the original vector and the vector after theand
parallel
transport.
fluxes F and q. The right-hand side of the Chern’s generMathematically it can be written like this:
alization of equation 3 is still an integer. Indeed, it is the
so-called Chern number. But there is a difference: It is not
necessarily an even integer, and g can no longer be interpreted as a tally of handles.
But if the Chern number doesn’t count handles, why
does it have to be an integer? One can see why by considering the
of parallel transport
around
the little
.
.
. loop
Where U and V are the vectors of length ε that form the parallelogram,
R failure
is Riemann
in figure 4. The loop can be thought of as the boundary of
Sphere:
K =
Torus:
∫
1
,
R2
dσ K = 4πR 2
∫
dσ K =
???
1
= 4π
R2
Returning to Laughlin’s argume
fundamental units, the Hall conduct
average charge transport in Laugh
ment are Chern numbers. That ex
ported charge, averaged over many
tized. And thus it explains the surpr
structure discovered in 1980 by von
The Hofstadter model
The glory of Chern numbers in th
from a theoretical model investigat
Hofstadter,9 three years before t
widely read book Gödel, Escher, Ba
Braid. The model, which was first
by Rudolph Peierls and his student
independent electrons on a 2D latt
mogeneous magnetic field. The mo
several reasons. First, its rich conje
perimentally realized in 2001 by C
Klitzing, and coworkers in a 2D el
lattice potential.10 The experiment v
detailed Hofstadter-model predictio
and coworkers.4 Second, the only kn
the Hall conductance in Hofstadter
numbers. Third, the model provide
built of Chern numbers.
The thermodynamic properties
in Hofstadter’s model are determine
the
. magnetic. flux, the temperature
.
choose a loop that is small enough around a point in a curved space, the amount of
change in the direction of a vector that is parallel transported along it is proportional
to the area enclosed by the loop. So, the ratio between the area of the loop and the
amount of change in the direction of the vector (whatever way we chose to measure
it) can be used as a measure to the curvature of the surface that includes the loop.
Actually we define curvature by the value of this ratio.
What about integrating over a closed surface?
Figure 4. The Gauss–Bonnet formula
trated here by a toroidal surface with
local curvature K is positive on those
face that resemble a sphere and nega
the hole, that resemble a saddle. Bec
of handles, equals one, the integral o
the entire surface vanishes. One can
Shiing-shen Chern’s quantum genera
Gauss–Bonnet formula plausible by c
angular mismatch of parallel transpo
around the small red patch in the fig
allel transport around the small loop enclosing the area
dA. The left side of the Gauss–Bonnet equation is geometric and not quantized a priori. But the right side is
manifestly quantized; the integer g is the number of handles characterizing the topology of S. (For the torus in figure 4, g = 1.) So, if we change K arbitrarily by denting the
Riemann Curvature Tensor
surface—so long as we do not punch through new handles—the quantized right side of equation 3 does not
Riemann tensor is a rank (1,3) tensor that describes the curvature in
all directions at a
change.
The
Gauss–Bonnet
given point in space. It takes 3 vectors and returns a single vector. The
vectors
that areformula has a well-known modern
generalization, due to Shiing-shen Chern. The Gaussfed to the tensor should be very small and have a length ε. If we
use the first
twoapplies not only to the geometry of
Bonnet-Chern
formula
surfaces
but
also
to the geometry of the eigenstates paravectors to form a tiny parallelogram and we parallel transport the meterized
third vector
around
by F and q. It looks exactly like equation 3, exthis parallelogram, the vector that the function returns is approximately
vector
cept that Kthe
is now
the adiabatic curvature of equation 1,
the surface
S is a torus parameterized by the two
difference between the original vector and the vector after theand
parallel
transport.
fluxes F and q. The right-hand side of the Chern’s generMathematically it can be written like this:
alization of equation 3 is still an integer. Indeed, it is the
so-called Chern number. But there is a difference: It is not
necessarily an even integer, and g can no longer be interpreted as a tally of handles.
But if the Chern number doesn’t count handles, why
does it have to be an integer? One can see why by considering the
of parallel transport
around
the little
.
.
. loop
Where U and V are the vectors of length ε that form the parallelogram,
R failure
is Riemann
in figure 4. The loop can be thought of as the boundary of
Sphere:
K =
Torus:
∫
1
,
R2
dσ K = 4πR 2
∫
dσ K =
???
1
= 4π
R2
Returning to Laughlin’s argume
fundamental units, the Hall conduct
average charge transport in Laugh
ment are Chern numbers. That ex
ported charge, averaged over many
tized. And thus it explains the surpr
structure discovered in 1980 by von
The Hofstadter model
The glory of Chern numbers in th
from a theoretical model investigat
Hofstadter,9 three years before t
widely read book Gödel, Escher, Ba
Braid. The model, which was first
by Rudolph Peierls and his student
independent electrons on a 2D latt
mogeneous magnetic field. The mo
several reasons. First, its rich conje
perimentally realized in 2001 by C
Klitzing, and coworkers in a 2D el
lattice potential.10 The experiment v
detailed Hofstadter-model predictio
and coworkers.4 Second, the only kn
the Hall conductance in Hofstadter
numbers. Third, the model provide
built of Chern numbers.
The thermodynamic properties
in Hofstadter’s model are determine
the
. magnetic. flux, the temperature
.
Grisha doesn't want it.
Emotiona
1 week ago
Gauss-Bonnet
theorem
(1848)
Separate from that cash,
Grisha has been
awarded the Fields Medal, equivalent
beginnin
SEOW!
to the Nobel Prize in math, which also comes with a nice clump of cash. (There
2 weeks ag
is no Nobel Prize in math.)
Over a smooth closed surface:
∫
Grisha doesn't want it.
1
dσ K = 2(1 − g)
2π S
Geometr
Geometric
Mounted S
4 weeks ag
The Gho
Tuesday F
start
Maybe his mama could talk some sense into this boy. But taking a look at this
Genus
g integer:
counts
theinto
number
ofprobably
“handles”
Rasputin
lookin'
mofo, if she could
talk sense
him, she'd
start by
4 weeks ag
g for
homeomorphic
notSame
dressin' him
funny
anymore.
surfaces
(continuous stretching and bending into a new shape)
Differentiability not needed
...other d
Court oka
spying
4 weeks ag
It's News
Shutting d
1 month ag
The Midd
Party (M
2 months a
Matty Boy, can you 'splain the Poincaré Conjecture to your gentle readers,
g=0
g=1
trinket99
World
Off With H
some of whom have serious issues with the math?
.
.
.
.
.
.
Grisha doesn't want it.
Emotiona
1 week ago
Gauss-Bonnet
theorem
(1848)
Separate from that cash,
Grisha has been
awarded the Fields Medal, equivalent
cée
gaz (8)
beginnin
SEOW!
to the Nobel Prize in math, which also comes with a nice clump of cash. (There
Math mugs Math humor Mathematician jokes Math mug Math mugs Math gift Math gifts
is no Nobel Prize in math.)
2 weeks ag
2/3/12 12:05 PM
Archimedes
Over a smooth closed surface:
∫
Grisha doesn't want it.
1
dσ K = 2(1 − g)
2π S
Geometr
Geometric
Mounted S
Eukleides (Euclid)
Descartes
Fermat
Pascal
Newton
Leibniz
Math Mug:
Certified Math Geek.
Euler
Lagrange
4 weeks ag
Laplace
Gauss
Ada Byron, Lady
Lovelace
The Gho
Tuesday F
start
Riemann
Maybe his mama could talk some sense into this boy. But taking a look at this
Cantor
Genus
g integer:
counts
theinto
number
ofprobably
“handles”
Rasputin
lookin'
mofo, if she could
talk sense
him, she'd
start by
surfaces
Qu'y-a-t-il
dans mon panier
...other d
Authors Pictures
Leonardo Da Vinci
Shop
Composers Posters
Composers T shirts
Famous Physicists
g for
homeomorphic
notSame
dressin' him
funny
anymore.
4 weeks ag
Math Mug (set theory):
If you consider the set of all sets that have
never been considered ....
(continuous stretching and bending into a new shape)
Produit(s)
Differentiability not Enlever
needed
Court oka
spying
4Lubian
weeks ag
cessoires (10)
potage
It's
News
blanche
Shutting
d
Math Mug - Topology
To a Topologist This is a Doughnut
1 month ag
The Midd
Party (M
2 months a
divers (19)Matty Boy, can you 'splain the Poincaré Conjecture to your gentle readers,
g=0
g=1
some of whom have serious issues with the math?
g=1
g=2
trinket99
World
Off With H
Math Mug:
Real Life is a Special Case
(black background)
http://mathematicianspictures.com/Math_Mugs_p01.htm
Page 4 of 6
.
.
.
.
.
.
Nonsmooth surfaces: Polyhedra
Euler characteristic
Euler characteristic - Wikipedia, the free encyclopedia
χ=V −E +F
2/3/12 5:27 PM
This result is known as Euler's polyhedron formula or theorem. It corresponds to the Euler characteristic
of the sphere (i.e. χ = 2), and applies identically to spherical polyhedra. An illustration of the formula on
some polyhedra is given below.
Name
Image
Toroidal Euler
polyhedroncharacteristic:
- Wikipedia, the free encyclopedia
Vertices Edges Faces
V
E
F
V−E+F
Tetrahedron
4
6
Hexahedron or cube
8
12
4 Wikipedia, the
2 free encyclopedia
From
In geometry, a toroidal polyhedron is a polyhedron with a genus of
1 or greater, representing topological torus surfaces.
6
2
Octahedron
6
12
Non-self-intersecting toroidal polyhedra are embedded tori, while
self-intersecting toroidal polyhedra are toroidal as abstract polyhedra,
which
can be verified
8
2 by their Euler characteristic (0 or less) and
orientability (orientable), and their self-intersecting realization in
Euclidean 3-space is a polyhedral immersion.
Dodecahedron
20
30
12
2
12
30
20
2
1 Stewart toroids
2 Császár and Szilassi polyhedra
3 Self-intersecting tori
The surfaces of nonconvex polyhedra can have various Euler4 characteristics;
See also
5 References
6 External
Vertices Edges Faces
Eulerlinks
characteristic:
Name
Image
V
E
F
A polyhedral torus can be constructed
to approximate a torus surface, from a
net of quadrilateral faces, like this 6x4
example.
χ=0
Contents
Icosahedron
2/3/12 5:34 PM
Toroidal polyhedron
V−E+F
χ = 2(1 − g)
Stewart toroids
Tetrahemihexahedron
6
12
7
1
A special category of toroidal polyhedra are constructed exclusively by
regular polygon faces, no intersections, and a further restriction that adjacent
. toroids, .
faces may not exist in the same plane. These are called Stewart
This toroidal polyhedron
.
constructed
from.
.
.
Nonsmooth surfaces: Polyhedra
Euler characteristic
Euler characteristic - Wikipedia, the free encyclopedia
χ=V −E +F
2/3/12 5:27 PM
This result is known as Euler's polyhedron formula or theorem. It corresponds to the Euler characteristic
of the sphere (i.e. χ = 2), and applies identically to spherical polyhedra. An illustration of the formula on
some polyhedra is given below.
Name
Image
Toroidal Euler
polyhedroncharacteristic:
- Wikipedia, the free encyclopedia
Vertices Edges Faces
V
E
F
V−E+F
Tetrahedron
4
6
Hexahedron or cube
8
12
4 Wikipedia, the
2 free encyclopedia
From
In geometry, a toroidal polyhedron is a polyhedron with a genus of
1 or greater, representing topological torus surfaces.
6
2
Octahedron
6
12
Non-self-intersecting toroidal polyhedra are embedded tori, while
self-intersecting toroidal polyhedra are toroidal as abstract polyhedra,
which
can be verified
8
2 by their Euler characteristic (0 or less) and
orientability (orientable), and their self-intersecting realization in
Euclidean 3-space is a polyhedral immersion.
Dodecahedron
20
30
12
2
12
30
20
2
1 Stewart toroids
2 Császár and Szilassi polyhedra
3 Self-intersecting tori
The surfaces of nonconvex polyhedra can have various Euler4 characteristics;
See also
5 References
6 External
Vertices Edges Faces
Eulerlinks
characteristic:
Name
Image
V
E
F
A polyhedral torus can be constructed
to approximate a torus surface, from a
net of quadrilateral faces, like this 6x4
example.
χ=0
Contents
Icosahedron
2/3/12 5:34 PM
Toroidal polyhedron
V−E+F
χ = 2(1 − g)
Stewart toroids
Tetrahemihexahedron
6
12
7
1
A special category of toroidal polyhedra are constructed exclusively by
regular polygon faces, no intersections, and a further restriction that adjacent
. toroids, .
faces may not exist in the same plane. These are called Stewart
This toroidal polyhedron
.
constructed
from.
.
.
Outline
1
What topology is about
2
Topology shows up in electronic structure
3
Classical Hall effect
4
2d noninteracting electrons in a magnetic field
5
Quantum Hall Effect
.
.
.
.
.
.
whereas the electrons in the Landau level give
rise to the same Hall current as that obtained
when all the electrons are in the level and can
move freely. Clearly this process must be occuring but its range of validity must be carefully
examined as an accompaniment to highly accurate
measurements of Hall resistance.
For high-precision measurements we used a
normal resistance R, in series with the device.
The voltage drop, U„across R„and the voltages
UH and Upp across and along the device was measured with a high impedance voltmeter (R &2 x10'0
corresponds to the minimum at V~ =8.7 V in Fig.
1, because the thicknesses of the gate oxides of
these two samples differ by a factor of 3.6. Our
experimental arrangement was not sensitive
enough to measure a value of R» of less than 0.1
0 which was found in the gate-voltage region
23.40 V& V &23.80 V. The Hall resistance in this
gate voltage region had a value of 6453.3+ 0.1 Q.
This inaccuracy of + 0.1 0 was due to the limited
sensitivity of the voltmeter. We would like to
mention that most of the samples, especially devices with a small length-to-width ratio, showed
a minimum in the Hall voltage as a function of V
at gate voltage close to the left side of the plateau.
In Fig. 2, this minimum is relatively shallow and
has a value of 6452. 87 0 at V~ =23.30 V.
In order to demonstrate the insensitivity of the
Hall resistance on the geometry of the device,
measurements on two samples with a length-towidth ratio of /W=0. 65 and I/W=25, respectively, are plotted in Fig. 3. The gate-voltage scale
Debut of topology in electronic structure (IQHE)
400
200.
I
23.0
23.5
24.5
24.0
=Vg /V
Figure from von Klitzing et al. (1980).
6454.
+
~
ow
z. 6453.
I
I
.0
—
+~)~+K «~~+
p
h/4e2
+
/
Gate voltage Vg was supposed to
control the carrier density.
II
6400-
1
I
8=13.0
T
1
I
T =1.8 K
(9
I
I
6452.
1
I
I
I
d
6300-
I
I
I
l
I
I
B=13.9 T
I
I
I
I
1
T
I
I
I
I
I
23.5
24.5
24.0
=Vg
/V
FIG. 2. Hall resistance RH, and device resistance,
Rpp, between the potential probes as a function of the
gate voltage ~~ in a region of gate voltage corresponding to a fully occupied, lowest (n =0) Landau level. The
plateau in RH has a value of 6453.3+ 0.1 Q. The geometry of the device was =400 pm, 8'=50 pm, and L»
I
I
I
-- ------L=260
I
23.0
0I
K
I
6451
6200
=1.8
pm, W=400pm
L=1000pm W=40 pm
0
Plateau flat to five decimal figures
I
6450
0.98
0.99
1.00
= gate
FIG. 3. Hall resistance
RH
101
voltage
102
Vg/r'el. units
for two samples with dif-
Natural
resistance
unit:
I
B=13 T.
&.
2
1 klitzing = h/e = 25812.807557(18) ohm.
This experiment: RH = klitzing / 4
=130 pm;
ferent geometry in a gate-voltage region V~ where the
n =0 Landau level is fully occupied. The recommended
value h/4e' is given as 6453.204
496
.
.
.
.
.
.
whereas the electrons in the Landau level give
rise to the same Hall current as that obtained
when all the electrons are in the level and can
move freely. Clearly this process must be occuring but its range of validity must be carefully
examined as an accompaniment to highly accurate
measurements of Hall resistance.
For high-precision measurements we used a
normal resistance R, in series with the device.
The voltage drop, U„across R„and the voltages
UH and Upp across and along the device was measured with a high impedance voltmeter (R &2 x10'0
corresponds to the minimum at V~ =8.7 V in Fig.
1, because the thicknesses of the gate oxides of
these two samples differ by a factor of 3.6. Our
experimental arrangement was not sensitive
enough to measure a value of R» of less than 0.1
0 which was found in the gate-voltage region
23.40 V& V &23.80 V. The Hall resistance in this
gate voltage region had a value of 6453.3+ 0.1 Q.
This inaccuracy of + 0.1 0 was due to the limited
sensitivity of the voltmeter. We would like to
mention that most of the samples, especially devices with a small length-to-width ratio, showed
a minimum in the Hall voltage as a function of V
at gate voltage close to the left side of the plateau.
In Fig. 2, this minimum is relatively shallow and
has a value of 6452. 87 0 at V~ =23.30 V.
In order to demonstrate the insensitivity of the
Hall resistance on the geometry of the device,
measurements on two samples with a length-towidth ratio of /W=0. 65 and I/W=25, respectively, are plotted in Fig. 3. The gate-voltage scale
Debut of topology in electronic structure (IQHE)
400
200.
I
23.0
23.5
24.5
24.0
=Vg /V
Figure from von Klitzing et al. (1980).
6454.
+
~
ow
z. 6453.
I
I
.0
—
+~)~+K «~~+
p
h/4e2
+
/
Gate voltage Vg was supposed to
control the carrier density.
II
6400-
1
I
8=13.0
T
1
I
T =1.8 K
(9
I
I
6452.
1
I
I
I
d
6300-
I
I
I
l
I
I
B=13.9 T
I
I
I
I
1
T
I
I
I
I
I
23.5
24.5
24.0
=Vg
/V
FIG. 2. Hall resistance RH, and device resistance,
Rpp, between the potential probes as a function of the
gate voltage ~~ in a region of gate voltage corresponding to a fully occupied, lowest (n =0) Landau level. The
plateau in RH has a value of 6453.3+ 0.1 Q. The geometry of the device was =400 pm, 8'=50 pm, and L»
I
I
I
-- ------L=260
I
23.0
0I
K
I
6451
6200
=1.8
pm, W=400pm
L=1000pm W=40 pm
0
Plateau flat to five decimal figures
I
6450
0.98
0.99
1.00
= gate
FIG. 3. Hall resistance
RH
101
voltage
102
Vg/r'el. units
for two samples with dif-
Natural
resistance
unit:
I
B=13 T.
&.
2
1 klitzing = h/e = 25812.807557(18) ohm.
This experiment: RH = klitzing / 4
=130 pm;
ferent geometry in a gate-voltage region V~ where the
n =0 Landau level is fully occupied. The recommended
value h/4e' is given as 6453.204
496
.
.
.
.
.
.
More recent experiments
Plateaus accurate to nine decimal figures
In the plateau regions ρxx = 0 and σxx = 0:
“quantum Hall insulator”
.
.
.
.
.
.
Outline
1
What topology is about
2
Topology shows up in electronic structure
3
Classical Hall effect
4
2d noninteracting electrons in a magnetic field
5
Quantum Hall Effect
.
.
.
.
.
.
Gaussian (a.k.a. CGS) units
Permittivity of free space ε0 =
1
4π
Permeability of free space µ0 = 4π
In vacuo D ≡ E and H ≡ B
All fields have the same dimensions
Newtonian & Hamiltonian mechanics:
(
)
1
dv
=f=Q E+ v×B
M
dt
c
H=
1
2M
(
p−
)2
Q
A(r) + QΦ(r)
c
.
.
.
.
.
.
Atomic Gaussian units
1
H=
2M
(
)2
1
p − A(r) + QΦ(r)
c
Schrödinger Hamiltonian for the electron
H=
)2
1 (
e
−i~∇ + A(r) − eΦ(r)
2me
c
me = 1, ~ = 1, e = 1,
(c = 137)
1 a.u. of energy = 1 hartree = 2 rydberg = 27.21 eV
1
H=
2
(
)2
1
−i∇ + A(r) − Φ(r)
c
Warning: Other “atomic units” with e =
.
√
2
.
.
.
.
.
Atomic Gaussian units
1
H=
2M
(
)2
1
p − A(r) + QΦ(r)
c
Schrödinger Hamiltonian for the electron
H=
)2
1 (
e
−i~∇ + A(r) − eΦ(r)
2me
c
me = 1, ~ = 1, e = 1,
(c = 137)
1 a.u. of energy = 1 hartree = 2 rydberg = 27.21 eV
1
H=
2
(
)2
1
−i∇ + A(r) − Φ(r)
c
Warning: Other “atomic units” with e =
.
√
2
.
.
.
.
.
Atomic Gaussian units
1
H=
2M
(
)2
1
p − A(r) + QΦ(r)
c
Schrödinger Hamiltonian for the electron
H=
)2
1 (
e
−i~∇ + A(r) − eΦ(r)
2me
c
me = 1, ~ = 1, e = 1,
(c = 137)
1 a.u. of energy = 1 hartree = 2 rydberg = 27.21 eV
1
H=
2
(
)2
1
−i∇ + A(r) − Φ(r)
c
Warning: Other “atomic units” with e =
.
√
2
.
.
.
.
.
Figure from Kittel ISSP, Ch. 6
.
.
.
.
.
.
Hall effect (1879)
2. Symmetry considerations
2D system
ric sketch
" j x # " ! xx
% j & $ % '!
( y ) ( xy
(E.R. Hall)
ffect (E.R. Hall)
" Ex #
udies of AHE
% E ...)
&$
Pugh, J. Smit,
(
y
)
! xy # " E x #
! xx &) %( E y &)
" * xx * xy # " j x #
% ' * Edwin*R. &Hall
% &
xx ) ( j y )
( xy
E (Karplus and Luttinger):
pin-orbit coupling
From Kittel ISSP (carriers of mass m and charge −e)
' * xy
*
! xx $ 2 xx 2
)
( ! xy $ * 2 +)* 2
(
* xx + * xy
xx
xy
"
1
1
with effect of scattering (Kohn, Luttinger) dv
+ v = −e E + v × ,B
m
H
dt
τ
c
mit)
* xx $
! xx
2
2
! xx
+ ! xy
* xy $
'! xy
2
2
+ ! xy
! xx
Steady-state:
!
dv
=0
dt
.
.
.
.
.
.
Drude-Zener theory
(
)
2. Symmetry considerations
eτ
1
v=−
E+ v×B
m
c
em
! xx ! xy # " E x #
! xy ! xx &) %( E y &)
* xx
' * xy
* xy # " j x #
* xx &) %( j y &)
* xx
2
2
x + * xy
! xx
2
2
xx + ! xy
In 2d, set Ey = 0;
! xy $
* xy $
cyclotron frequency
' * xy
2
* xx2 + * xy
'! xy
2
2
+ ! xy
! xx
ωc =
eB
mc
"
vx
vy
eτ
= − Ex,H− ωc τ vy
m
= ωc τ vx
.
!
.
.
.
.
.
Hall conductivity
Current
j = −ne v
(n carrier density)
jx
jy
ne2 τ
Ex − ωc τ jy
m
= ωc τ jx
=
In zero B field
jx = σ0 Ex ,
σ0 =
ne2 τ
m
In a B field
jx
=
jy
=
σ0
Ex = σxx Ex
1 + (ωc τ )2
ωc τ σ 0
Ex = σyx Ex
1 + (ωc τ )2
.
.
.
.
.
.
Conductivity vs. resistivity (classical & quantum)
(
jx
jy
)
(
=
σxx
σyx
↔
−σyx
σyy
)(
Ex
Ey
)
↔
ρ = ( σ )−1
ρxx =
At B = 0
2
σxx
σxx
,
2
+ σyx
ρxy =
2
σxx
σyx
2
+ σyx
ρxx = 1/σxx
In the nondissipative regime (j · E = 0)
σxx = 0
and
ρxx = 0
ρxy = 1/σyx
.
.
.
.
.
.
Conductivity vs. resistivity (classical & quantum)
(
jx
jy
)
(
=
σxx
σyx
↔
−σyx
σyy
)(
Ex
Ey
)
↔
ρ = ( σ )−1
ρxx =
At B = 0
2
σxx
σxx
,
2
+ σyx
ρxy =
2
σxx
σyx
2
+ σyx
ρxx = 1/σxx
In the nondissipative regime (j · E = 0)
σxx = 0
and
ρxx = 0
ρxy = 1/σyx
.
.
.
.
.
.
Conductivity vs. resistivity (classical & quantum)
(
jx
jy
)
(
=
σxx
σyx
↔
−σyx
σyy
)(
Ex
Ey
)
↔
ρ = ( σ )−1
ρxx =
At B = 0
2
σxx
σxx
,
2
+ σyx
ρxy =
2
σxx
σyx
2
+ σyx
ρxx = 1/σxx
In the nondissipative regime (j · E = 0)
σxx = 0
and
ρxx = 0
ρxy = 1/σyx
.
.
.
.
.
.
Nondissipative limit (τ → ∞, classical Drude-Zener)
σ0 =
ne2 τ
m
At B = 0
σxx =
σxx = σ0
At B ̸= 0
for
ωc τ σ 0
1 + (ωc τ )2
σyx =
diverges
τ ≫ 1/ωc
σxx
= 0,
ρxy
= 1/σyx
=
σ0
1 + (ωc τ )2
1
B
nec
ρxx = 0
(longitudinal resistivity)
mωc
m eB
=
=
ne2
ne2 mc
(Hall resistivity)
.
.
.
.
.
.
Nondissipative limit (τ → ∞, classical Drude-Zener)
σ0 =
ne2 τ
m
At B = 0
σxx =
σxx = σ0
At B ̸= 0
for
ωc τ σ 0
1 + (ωc τ )2
σyx =
diverges
τ ≫ 1/ωc
σxx
= 0,
ρxy
= 1/σyx
=
σ0
1 + (ωc τ )2
1
B
nec
ρxx = 0
(longitudinal resistivity)
mωc
m eB
=
=
ne2
ne2 mc
(Hall resistivity)
.
.
.
.
.
.
Nondissipative limit (τ → ∞, classical Drude-Zener)
σ0 =
ne2 τ
m
At B = 0
σxx =
σxx = σ0
At B ̸= 0
for
ωc τ σ 0
1 + (ωc τ )2
σyx =
diverges
τ ≫ 1/ωc
σxx
= 0,
ρxy
= 1/σyx
=
σ0
1 + (ωc τ )2
1
B
nec
ρxx = 0
(longitudinal resistivity)
mωc
m eB
=
=
ne2
ne2 mc
(Hall resistivity)
.
.
.
.
.
.
Multiplying and dividing by h
In 2d resistance/resistivity and conductance/conductivity
have the same dimensions: do they coincide?
n = N/A
(number of carrriers per unit area)
1
AB
Φ
1 h
B=
=
=
nec
Nec
Nec
ν e2
Φ magnetic flux through area A
h/e2 ≃ 25813 Ω (natural resistance unit)
ν dimensionless
ρxy =
ν=
NΦ0
Φ
filling factor,
Φ0 =
hc
e
flux quantum
ν = (number of electrons)/(number of flux quanta)
.
.
.
.
.
.
Multiplying and dividing by h
In 2d resistance/resistivity and conductance/conductivity
have the same dimensions: do they coincide?
n = N/A
(number of carrriers per unit area)
1
AB
Φ
1 h
B=
=
=
nec
Nec
Nec
ν e2
Φ magnetic flux through area A
h/e2 ≃ 25813 Ω (natural resistance unit)
ν dimensionless
ρxy =
ν=
NΦ0
Φ
filling factor,
Φ0 =
hc
e
flux quantum
ν = (number of electrons)/(number of flux quanta)
.
.
.
.
.
.
Multiplying and dividing by h
In 2d resistance/resistivity and conductance/conductivity
have the same dimensions: do they coincide?
n = N/A
(number of carrriers per unit area)
1
AB
Φ
1 h
B=
=
=
nec
Nec
Nec
ν e2
Φ magnetic flux through area A
h/e2 ≃ 25813 Ω (natural resistance unit)
ν dimensionless
ρxy =
ν=
NΦ0
Φ
filling factor,
Φ0 =
hc
e
flux quantum
ν = (number of electrons)/(number of flux quanta)
.
.
.
.
.
.
Multiplying and dividing by h
In 2d resistance/resistivity and conductance/conductivity
have the same dimensions: do they coincide?
n = N/A
(number of carrriers per unit area)
1
AB
Φ
1 h
B=
=
=
nec
Nec
Nec
ν e2
Φ magnetic flux through area A
h/e2 ≃ 25813 Ω (natural resistance unit)
ν dimensionless
ρxy =
ν=
NΦ0
Φ
filling factor,
Φ0 =
hc
e
flux quantum
ν = (number of electrons)/(number of flux quanta)
.
.
.
.
.
.
A typical experiment
GaAs-GaAlAs heterojunction, at 30mK
.
.
.
.
.
.
Outline
1
What topology is about
2
Topology shows up in electronic structure
3
Classical Hall effect
4
2d noninteracting electrons in a magnetic field
5
Quantum Hall Effect
.
.
.
.
.
.
Hamiltonian in B field (flat substrate potential)
N noninteracting (& spin-polarized) electrons in zero potential:
N
]2
1 ∑[
e
Ĥ =
pi + A(ri )
2me
c
i=1
Gaussian units
me electron mass
−e electron charge
(
)
1
e
me pi + c A(ri ) velocity
pi = −i~∇i canonical momentum
B = ∇ × A(r)
.
.
.
.
.
.
Landau gauge
Everything in 2d;
B uniform, along z.
Ax = 0,
Ay = Bx
For each electron the Hamiltonian is
[
(
)2 ]
∂
~2
∂2
e
H(x, y ) =
− 2 + −i
+
Bx
2me
∂y
~c
∂x
Landau ansatz ψk (x, y ) = eiky φk (x)
~2 iky ′′
~2
−
e φk (x) +
2me
2me
(
eB
k+
x
~c
)2
eiky φk (x) = εk eiky φk (x).
Harmonic oscillator in 1d
.
.
.
.
.
.
Landau gauge
Everything in 2d;
B uniform, along z.
Ax = 0,
Ay = Bx
For each electron the Hamiltonian is
[
(
)2 ]
∂
~2
∂2
e
H(x, y ) =
− 2 + −i
+
Bx
2me
∂y
~c
∂x
Landau ansatz ψk (x, y ) = eiky φk (x)
~2 iky ′′
~2
−
e φk (x) +
2me
2me
(
eB
k+
x
~c
)2
eiky φk (x) = εk eiky φk (x).
Harmonic oscillator in 1d
.
.
.
.
.
.
Landau gauge
Everything in 2d;
B uniform, along z.
Ax = 0,
Ay = Bx
For each electron the Hamiltonian is
[
(
)2 ]
∂
~2
∂2
e
H(x, y ) =
− 2 + −i
+
Bx
2me
∂y
~c
∂x
Landau ansatz ψk (x, y ) = eiky φk (x)
~2 iky ′′
~2
−
e φk (x) +
2me
2me
(
eB
k+
x
~c
)2
eiky φk (x) = εk eiky φk (x).
Harmonic oscillator in 1d
.
.
.
.
.
.
Landau oscillator
(
)
~2 ′′
~2
eB 2
φk (x) +
k+
x φk (x) = εk φk (x)
2me
2me
~c
(
)
(
)2
2
~2 ′′
1
eB
~c
−
φ (x) + me
x+
k φk (x) = εk φk (x)
2me k
2
me c
eB
−
Harmonic oscillator
~c
Center in xk = − eB
k = −ℓ2 k
ℓ = (~c/eB)1/2 “magnetic length” (diverges for B → 0)
Frequency ωc = meBe c
cyclotron frequency
(classical, Gaussian units)
.
.
.
.
.
.
Landau oscillator
(
)
~2 ′′
~2
eB 2
φk (x) +
k+
x φk (x) = εk φk (x)
2me
2me
~c
(
)
(
)2
2
~2 ′′
1
eB
~c
−
φ (x) + me
x+
k φk (x) = εk φk (x)
2me k
2
me c
eB
−
Harmonic oscillator
~c
Center in xk = − eB
k = −ℓ2 k
ℓ = (~c/eB)1/2 “magnetic length” (diverges for B → 0)
Frequency ωc = meBe c
cyclotron frequency
(classical, Gaussian units)
.
.
.
.
.
.
Landau oscillator
(
)
~2 ′′
~2
eB 2
φk (x) +
k+
x φk (x) = εk φk (x)
2me
2me
~c
(
)
(
)2
2
~2 ′′
1
eB
~c
−
φ (x) + me
x+
k φk (x) = εk φk (x)
2me k
2
me c
eB
−
Harmonic oscillator
~c
Center in xk = − eB
k = −ℓ2 k
ℓ = (~c/eB)1/2 “magnetic length” (diverges for B → 0)
Frequency ωc = meBe c
cyclotron frequency
(classical, Gaussian units)
.
.
.
.
.
.
Eigenvalues and eigenvectors
Spectrum independent of k :
εn = (n + 12 )ωc
Ground-state orbitals (LLL):
ψk (x, y ) = eiky φk (x) = eiky χ(x + ℓ2 k )
(
χ(x) =
1
πℓ2
)1/4
e−x
2 /(2ℓ2 )
Infinite degeneracy: one orbital for each k
Electron confined in a vertical strip centered at ℓ2 k
What about the current?
Any unitary transformation of the LLL orbitals is an
eigenfunction
.
.
.
.
.
.
Eigenvalues and eigenvectors
Spectrum independent of k :
εn = (n + 12 )ωc
Ground-state orbitals (LLL):
ψk (x, y ) = eiky φk (x) = eiky χ(x + ℓ2 k )
(
χ(x) =
1
πℓ2
)1/4
e−x
2 /(2ℓ2 )
Infinite degeneracy: one orbital for each k
Electron confined in a vertical strip centered at ℓ2 k
What about the current?
Any unitary transformation of the LLL orbitals is an
eigenfunction
.
.
.
.
.
.
Eigenvalues and eigenvectors
Spectrum independent of k :
εn = (n + 12 )ωc
Ground-state orbitals (LLL):
ψk (x, y ) = eiky φk (x) = eiky χ(x + ℓ2 k )
(
χ(x) =
1
πℓ2
)1/4
e−x
2 /(2ℓ2 )
Infinite degeneracy: one orbital for each k
Electron confined in a vertical strip centered at ℓ2 k
What about the current?
Any unitary transformation of the LLL orbitals is an
eigenfunction
.
.
.
.
.
.
Eigenvalues and eigenvectors
Spectrum independent of k :
εn = (n + 12 )ωc
Ground-state orbitals (LLL):
ψk (x, y ) = eiky φk (x) = eiky χ(x + ℓ2 k )
(
χ(x) =
1
πℓ2
)1/4
e−x
2 /(2ℓ2 )
Infinite degeneracy: one orbital for each k
Electron confined in a vertical strip centered at ℓ2 k
What about the current?
Any unitary transformation of the LLL orbitals is an
eigenfunction
.
.
.
.
.
.
Counting the states (discretize k)
(
iky
ψk (x, y ) = e
χ(x − ℓ k )
2
χ(x) =
1
πℓ2
)1/4
e−x
ki+1 − ki =
Periodic boundary conditions in y :
2 /(2ℓ2 )
2π
L
2πℓ2
L
Horizontal distance between neighboring orbitals:
Area covered by one state: 2πℓ2
Number of states in each LL: N =
A
2πℓ2
Magnetic flux:
hc
Φ = AB = N 2πℓ2 B = N 2π~c
e = N e = N Φ0
Flux quantum: Φ0 =
hc
e
(Φ0 =
h
e
in SI units)
Φ0 a universal constant
.
.
.
.
.
.
Counting the states (discretize k)
(
iky
ψk (x, y ) = e
χ(x − ℓ k )
2
χ(x) =
1
πℓ2
)1/4
e−x
ki+1 − ki =
Periodic boundary conditions in y :
2 /(2ℓ2 )
2π
L
2πℓ2
L
Horizontal distance between neighboring orbitals:
Area covered by one state: 2πℓ2
Number of states in each LL: N =
A
2πℓ2
Magnetic flux:
hc
Φ = AB = N 2πℓ2 B = N 2π~c
e = N e = N Φ0
Flux quantum: Φ0 =
hc
e
(Φ0 =
h
e
in SI units)
Φ0 a universal constant
.
.
.
.
.
.
Density of states
At B = 0:
At B ̸= 0:
eA
D(ε) = constant = 2πm
h2
Φ/Φ0 states in each LL
(
)
∞
Φ ∑
1
D(ε) =
δ ε − (n + )~ωc
Φ0
2
n=1
maximum filling for each LL is ν = 1.
.
.
.
.
.
.
Density of states
At B = 0:
At B ̸= 0:
eA
D(ε) = constant = 2πm
h2
Φ/Φ0 states in each LL
(
)
∞
Φ ∑
1
D(ε) =
δ ε − (n + )~ωc
Φ0
2
n=1
maximum filling for each LL is ν = 1.
.
.
.
.
.
.
Density of states
How many states in the hatched region?
∫
ε
ε+~ωc
dε′ D(ε′ ) = ~ωc
2πme A
Φ
=
2
Φ0
h
.
.
.
.
.
.
Outline
1
What topology is about
2
Topology shows up in electronic structure
3
Classical Hall effect
4
2d noninteracting electrons in a magnetic field
5
Quantum Hall Effect
.
.
.
.
.
.
What the experiment shows
In modern jargon: The plateaus are “topologically protected”
.
.
.
.
.
.
Wavefunction “knotted” or “twisted”
PERSPECTIVE INSIGHT
NATURE|Vol 464|11 March 2010
a
b
Figure 1 | Metallic states are born when a surface unties ‘knotted’ electron
wavefunctions. a, An illustration of topological change and the resultant
surface state. The trefoil knot (left) and the simple loop (right) represent
different insulating materials: the knot is a topological insulator, and the
loop is an ordinary insulator. Because there is no continuous deformation
by which one can be converted into the other, there must be a surface where
the string is cut, shown as a string with open ends (centre), to pass between
the two knots; more formally, the topological invariants cannot remain
defined. If the topological invariants are always defined for an insulator,
then the surface must be metallic. b, The simplest example of a knotted 3D
electronic band structure (with two bands)35, known to mathematicians as
the Hopf map. The full topological structure would also have linked fibres
on each ring, in addition to the linking of rings shown here. The knotting
in real topological insulators is more complex as these require a minimum
of four electronic bands, but the surface structure that appears is relatively
simple (Fig. 3).
introduced in around 2003 it can lead to a quantum spin Hall effect,
in which electrons with opposite spin angular momentum (commonly
called spin up and spin down) move in opposite directions around the
edge of the droplet in the absence of an external magnetic field2 (Fig. 2b).
These simplified models were the first steps towards understanding
topological insulators. But it was unclear how realistic the models were:
in real materials, there is mixing of spin-up and spin-down electrons,
of the quantum spin Hall effect (discussed earlier), which may allow 2D
Knotted in reciprocal space in nontrivial ways
The famous TKNN paper:
D. J. Thouless, M. Kohmoto, M. P. Nightingale, and M. den
Nijs, Phys. Rev. Lett. 49, 405 (1982).
symmetry as an applied magnetic field would), in simplified models in the crystal) will always contain a quantum wire like that at the edge
topological insulator physics to be observed in a 3D material .
Integer numbers are very “robust”
9
There is also, however, a ‘strong’ topological insulator, which has a
more subtle relationship to the 2D case; the relationship is that in two
dimensions it is possible to connect ordinary insulators and topological insulators smoothly by breaking time-reversal symmetry7. Such a
continuous interpolation
can be. used to build
a 3D. band structure
that
.
.
.
.
Is it possible to “cut the knots”?
Wavefunction “knotted” or “twisted”
PERSPECTIVE INSIGHT
NATURE|Vol 464|11 March 2010
a
b
Figure 1 | Metallic states are born when a surface unties ‘knotted’ electron
wavefunctions. a, An illustration of topological change and the resultant
surface state. The trefoil knot (left) and the simple loop (right) represent
different insulating materials: the knot is a topological insulator, and the
loop is an ordinary insulator. Because there is no continuous deformation
by which one can be converted into the other, there must be a surface where
the string is cut, shown as a string with open ends (centre), to pass between
the two knots; more formally, the topological invariants cannot remain
defined. If the topological invariants are always defined for an insulator,
then the surface must be metallic. b, The simplest example of a knotted 3D
electronic band structure (with two bands)35, known to mathematicians as
the Hopf map. The full topological structure would also have linked fibres
on each ring, in addition to the linking of rings shown here. The knotting
in real topological insulators is more complex as these require a minimum
of four electronic bands, but the surface structure that appears is relatively
simple (Fig. 3).
introduced in around 2003 it can lead to a quantum spin Hall effect,
in which electrons with opposite spin angular momentum (commonly
called spin up and spin down) move in opposite directions around the
edge of the droplet in the absence of an external magnetic field2 (Fig. 2b).
These simplified models were the first steps towards understanding
topological insulators. But it was unclear how realistic the models were:
in real materials, there is mixing of spin-up and spin-down electrons,
of the quantum spin Hall effect (discussed earlier), which may allow 2D
Knotted in reciprocal space in nontrivial ways
The famous TKNN paper:
D. J. Thouless, M. Kohmoto, M. P. Nightingale, and M. den
Nijs, Phys. Rev. Lett. 49, 405 (1982).
symmetry as an applied magnetic field would), in simplified models in the crystal) will always contain a quantum wire like that at the edge
topological insulator physics to be observed in a 3D material .
Integer numbers are very “robust”
9
There is also, however, a ‘strong’ topological insulator, which has a
more subtle relationship to the 2D case; the relationship is that in two
dimensions it is possible to connect ordinary insulators and topological insulators smoothly by breaking time-reversal symmetry7. Such a
continuous interpolation
can be. used to build
a 3D. band structure
that
.
.
.
.
Is it possible to “cut the knots”?
Quantum Mechanics:
Data for a torus
Berry connection & Berry curvature
Parametric Hamiltonian What data fully
specify homogeneous
on a closed surface (a torus)
:
magnetic field on torus?
H(ϑ, φ) = H(ϑ + 2π, φ) = H(ϑ, φ + 2π)
Ground nondegenerate eigenstate |ψ0 (ϑ, φ)⟩
Field, ,two fundamental loops and two fluxes
Berry connection:
Aϑ (ϑ, φ) = i ⟨ψ0 |
∂
ψ0 ⟩,
∂ϑ
Aφ (ϑ, φ) = i ⟨ψ0 |
∂
Physical information (observable)
∂φ
ψ0 ⟩
Berry curvature:
(
Ω(ϑ, φ) = i
)
∂
∂
∂
∂
⟨ ψ0 | ψ0 ⟩ − ⟨ ψ0 | ψ0 ⟩
∂ϑ
∂φ
∂φ
∂ϑ
.
.
.
.
.
.
Data for a torus
A flavor of the theory: Gauss-Bonnet-Chern theorem
at data fully
y homogeneous
c field on torus?
Berry curvature:
two fundamental loops and two fluxes
(
Ω(ϑ, φ) = i
)
∂
∂
∂
∂
⟨ ψ0 | ψ0 ⟩ − ⟨ ψ0 | ψ0 ⟩
∂ϑ
∂φ
∂φ
∂ϑ
Chern number (a.k.a. TKNN invariant):
∫ 2π
Physical information ∫
(observable)
2π
1
2π
dφ Ω(ϑ, φ) = C ∈ Z
dϑ
0
0
Analogue of the genus in differential geometry
.
.
.
.
.
.
Data for a torus
A flavor of the theory: Gauss-Bonnet-Chern theorem
at data fully
y homogeneous
c field on torus?
Berry curvature:
two fundamental loops and two fluxes
(
Ω(ϑ, φ) = i
)
∂
∂
∂
∂
⟨ ψ0 | ψ0 ⟩ − ⟨ ψ0 | ψ0 ⟩
∂ϑ
∂φ
∂φ
∂ϑ
Chern number (a.k.a. TKNN invariant):
∫ 2π
Physical information ∫
(observable)
2π
1
2π
dφ Ω(ϑ, φ) = C ∈ Z
dϑ
0
0
Analogue of the genus in differential geometry
.
.
.
.
.
.
Role of disorder
Current carried by delocalized states only
.
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