Download The Integer Quantum Hall Effect

The Integer Quantum Hall E↵ect
Rhine Samajdar
Harvard University
December 14, 2016
Rhine Samajdar (Harvard University)
The Integer Quantum Hall E↵ect
December 14, 2016
1 / 41
Introduction
Table of Contents
1
Introduction
The Classical Hall E↵ect
Integer Quantum Hall E↵ect
2
Landau Levels
Review
Conductivity: A quick
calculation
3
5
6
Büttiker’s Theory
Edge modes
Disorder and Localization
Rhine Samajdar (Harvard University)
4
7
Laughlin’s Gedankenexperiment
Gauge Invariance
Spectral Flow
Topology
Linear Response
Quantized Hall Conductivity
Particles on a Lattice
The TKNN formula
Peierls substitution
Chern Numbers
References
The Integer Quantum Hall E↵ect
December 14, 2016
2 / 41
Introduction
The Classical Hall E↵ect
The Classical Hall E↵ect
A magnetic field causes charged particles to move in circles. Let
B = (0, 0, B), E = (E, 0, 0),
v = (ẋ, ẏ, 0 ).
Figure: The setup for the Hall e↵ect.
Rhine Samajdar (Harvard University)
The Integer Quantum Hall E↵ect
December 14, 2016
3 / 41
Introduction
The Classical Hall E↵ect
The Classical Hall E↵ect
A magnetic field causes charged particles to move in circles. Let
B = (0, 0, B), E = (E, 0, 0),
v = (ẋ, ẏ, 0 ).
Figure: The setup for the Hall e↵ect.
Rhine Samajdar (Harvard University)
Edwin Hall, 1879
In equilibrium, a current in the xdirection requires an electric field with
a component in the y-direction!
The Integer Quantum Hall E↵ect
December 14, 2016
3 / 41
Introduction
The Classical Hall E↵ect
The Classical Hall E↵ect
Drude model:
m
dv
=
dt
eE
ev ⇥ B
Rhine Samajdar (Harvard University)
mv
⌧
The Integer Quantum Hall E↵ect
December 14, 2016
4 / 41
Introduction
The Classical Hall E↵ect
The Classical Hall E↵ect
Drude model:
m
dv
=
dt
J=
=
1
✓
ne2 ⌧
m
2
+ !B
xx
xy
xy
⌧2
mv
⌧
ev ⇥ B
eE
✓
yy
1
!B ⌧
Rhine Samajdar (Harvard University)
◆
E
!B ⌧
1
◆
The Integer Quantum Hall E↵ect
December 14, 2016
4 / 41
Introduction
The Classical Hall E↵ect
The Classical Hall E↵ect
Drude model:
m
dv
=
dt
J=
=
1
Hall coefficient:
✓
ne2 ⌧
m
2
+ !B
xx
xy
xy
⌧2
mv
⌧
ev ⇥ B
eE
✓
yy
1
!B ⌧
Rhine Samajdar (Harvard University)
◆
E
!B ⌧
1
Ey
⇢xy
=
Jx B
B
!B
1
=
=
B DC
ne
RH =
◆
The Integer Quantum Hall E↵ect
December 14, 2016
4 / 41
Introduction
The Classical Hall E↵ect
The Classical Hall E↵ect
Drude model:
m
dv
=
dt
J=
=
1
Hall coefficient:
✓
ne2 ⌧
m
2
+ !B
xx
xy
xy
⌧2
mv
⌧
ev ⇥ B
eE
✓
yy
1
!B ⌧
Rhine Samajdar (Harvard University)
◆
Ey
⇢xy
=
Jx B
B
!B
1
=
=
B DC
ne
RH =
E
!B ⌧
1
◆
⇢xx =
The Integer Quantum Hall E↵ect
m
;
n e2 ⌧
⇢xy =
B
ne
December 14, 2016
4 / 41
Introduction
Integer Quantum Hall E↵ect
The Integer Quantum Hall E↵ect
Figure: Klitzing, Dorda, and Pepper,
PRL 45, 494 (1980).
Rhine Samajdar (Harvard University)
The Integer Quantum Hall E↵ect
December 14, 2016
5 / 41
Introduction
Integer Quantum Hall E↵ect
The Integer Quantum Hall E↵ect
The Hall resistivity ⇢xy sits on a
plateau for a range of magnetic
fields. On these plateaus,
⇢xy = eh2 ⌫1 ; ⌫ 2 Z.
Figure: Klitzing, Dorda, and Pepper,
PRL 45, 494 (1980).
Rhine Samajdar (Harvard University)
The Integer Quantum Hall E↵ect
December 14, 2016
5 / 41
Introduction
Integer Quantum Hall E↵ect
The Integer Quantum Hall E↵ect
The Hall resistivity ⇢xy sits on a
plateau for a range of magnetic
fields. On these plateaus,
⇢xy = eh2 ⌫1 ; ⌫ 2 Z.
When ⇢xy sits on a plateau, the
longitudinal resistivity vanishes:
⇢xx = 0.
Figure: Klitzing, Dorda, and Pepper,
PRL 45, 494 (1980).
Rhine Samajdar (Harvard University)
The Integer Quantum Hall E↵ect
December 14, 2016
5 / 41
Introduction
Integer Quantum Hall E↵ect
The Integer Quantum Hall E↵ect
The Hall resistivity ⇢xy sits on a
plateau for a range of magnetic
fields. On these plateaus,
⇢xy = eh2 ⌫1 ; ⌫ 2 Z.
When ⇢xy sits on a plateau, the
longitudinal resistivity vanishes:
⇢xx = 0.
Figure: Klitzing, Dorda, and Pepper,
PRL 45, 494 (1980).
Rhine Samajdar (Harvard University)
The center of each of plateau
h
occurs at B = n⌫ 0 ;
0 = e.
The Integer Quantum Hall E↵ect
December 14, 2016
5 / 41
Landau Levels
Table of Contents
1
Introduction
The Classical Hall E↵ect
Integer Quantum Hall E↵ect
2
Landau Levels
Review
Conductivity: A quick
calculation
3
5
6
Büttiker’s Theory
Edge modes
Disorder and Localization
Rhine Samajdar (Harvard University)
4
7
Laughlin’s Gedankenexperiment
Gauge Invariance
Spectral Flow
Topology
Linear Response
Quantized Hall Conductivity
Particles on a Lattice
The TKNN formula
Peierls substitution
Chern Numbers
References
The Integer Quantum Hall E↵ect
December 14, 2016
6 / 41
Landau Levels
Review
Landau Levels
Landau gauge: A = x B ŷ
n,k (x, y)
Symmetric gauge: A =
2
⇠ eiky Hn (x + k lB
)
2
2
⇥ e (x+k lB )/2lB
The wavefunctions look like strips,
extended along ŷ but exponentially
2 in the x̂
localized around x = klB
direction.
Rhine Samajdar (Harvard University)
LLL,m
⇠
✓
z
lB
◆m
e
1
2r
⇥B
2
|z|2 /4lB
The wavefunction with angular momentum mpis peaked on a ring of
radius r = 2mlB .
The Integer Quantum Hall E↵ect
December 14, 2016
7 / 41
Landau Levels
Review
Landau Levels
Landau gauge: A = x B ŷ
n,k (x, y)
Symmetric gauge: A =
2
⇠ eiky Hn (x + k lB
)
2
LLL,m
2
⇠
✓
z
lB
◆m
e
1
2r
⇥B
2
|z|2 /4lB
⇥ e (x+k lB )/2lB
The wavefunctions look like strips, The wavefunction with angular moextended along ŷ but exponentially mentum mpis peaked on a ring of
2 in the x̂ radius r =
localized around x = klB
2mlB .
direction.
✓
◆
1
AB
En = ~ ! B n +
; Degeneracy: N =
2
0
Rhine Samajdar (Harvard University)
The Integer Quantum Hall E↵ect
December 14, 2016
7 / 41
Landau Levels
Review
Landau Levels
Figure: Landau level wavefunctions in the symmetric gauge (Figure courtesy Emil
J. Bergholtz).
Rhine Samajdar (Harvard University)
The Integer Quantum Hall E↵ect
December 14, 2016
8 / 41
Landau Levels
Conductivity: A quick calculation
Conductivity in Filled Landau Levels
⇢xy =
h 1
e2 ⌫
Rhine Samajdar (Harvard University)
The Integer Quantum Hall E↵ect
December 14, 2016
9 / 41
Landau Levels
Conductivity: A quick calculation
Conductivity in Filled Landau Levels
⇢xy =
h 1
e2 ⌫
Rhine Samajdar (Harvard University)
⌫ Landau levels are filled.
The Integer Quantum Hall E↵ect
December 14, 2016
9 / 41
Landau Levels
Conductivity: A quick calculation
Ansatz: Drude model
On a plateau, the Hall resistivity takes the value ⇢xy =
Rhine Samajdar (Harvard University)
The Integer Quantum Hall E↵ect
h 1
.
e2 ⌫
December 14, 2016
10 / 41
Landau Levels
Conductivity: A quick calculation
Ansatz: Drude model
h 1
.
e2 ⌫
B
⇢xy = n e .
On a plateau, the Hall resistivity takes the value ⇢xy =
From our classical calculation in the Drude model
Rhine Samajdar (Harvard University)
The Integer Quantum Hall E↵ect
December 14, 2016
10 / 41
Landau Levels
Conductivity: A quick calculation
Ansatz: Drude model
h 1
.
e2 ⌫
B
⇢xy = n e .
B
On a plateau, the Hall resistivity takes the value ⇢xy =
From our classical calculation in the Drude model
Thus, to get the resistivity of the ⌫ th plateau, n = 0 ⌫.
This is exactly the density of electrons required to fill ⌫ Landau levels!
Rhine Samajdar (Harvard University)
The Integer Quantum Hall E↵ect
December 14, 2016
10 / 41
Landau Levels
Conductivity: A quick calculation
Ansatz: Drude model
h 1
.
e2 ⌫
B
⇢xy = n e .
B
On a plateau, the Hall resistivity takes the value ⇢xy =
From our classical calculation in the Drude model
Thus, to get the resistivity of the ⌫ th plateau, n = 0 ⌫.
This is exactly the density of electrons required to fill ⌫ Landau levels!
Further, when ⌫ Landau levels are filled, there is a gap in the energy spectrum. When we turn on a small electric field, the electrons are stuck in
place like in an insulator. This means that the scattering time ⌧ ! 1 and
⇢xx = 0.
Rhine Samajdar (Harvard University)
The Integer Quantum Hall E↵ect
December 14, 2016
10 / 41
Büttiker’s Theory
Table of Contents
1
Introduction
The Classical Hall E↵ect
Integer Quantum Hall E↵ect
2
Landau Levels
Review
Conductivity: A quick
calculation
3
5
6
Büttiker’s Theory
Edge modes
Disorder and Localization
Rhine Samajdar (Harvard University)
4
7
Laughlin’s Gedankenexperiment
Gauge Invariance
Spectral Flow
Topology
Linear Response
Quantized Hall Conductivity
Particles on a Lattice
The TKNN formula
Peierls substitution
Chern Numbers
References
The Integer Quantum Hall E↵ect
December 14, 2016
11 / 41
Büttiker’s Theory
Conductivity for a single free particle
Particle velocity: m ẋ = p + e A
Rhine Samajdar (Harvard University)
The Integer Quantum Hall E↵ect
December 14, 2016
12 / 41
Büttiker’s Theory
Conductivity for a single free particle
Particle velocity: m ẋ = p + e A
e P
Total current: I = m
filledstates h | i ~r + eA| i
Rhine Samajdar (Harvard University)
The Integer Quantum Hall E↵ect
December 14, 2016
12 / 41
Büttiker’s Theory
Conductivity for a single free particle
Particle velocity: m ẋ = p + e A
e P
Total current: I = m
filledstates h | i ~r + eA| i
Thus, with ⌫ Landau levels filled:
⌫
Ix =
Iy =
e XX
m
e
m
⌧
n=1 k
⌫ X⌧
X
n=1 k
Rhine Samajdar (Harvard University)
n,k
i~
@
@x
n,k
i~
@
+ exB
@y
=0
n,k
The Integer Quantum Hall E↵ect
n,k
=
e⌫
XE
k
B
December 14, 2016
12 / 41
Büttiker’s Theory
Conductivity for a single free particle
Hence,
✓ ◆
✓
E
0
E=
)J=
0
e ⌫ E/
Rhine Samajdar (Harvard University)
The Integer Quantum Hall E↵ect
0
◆
December 14, 2016
13 / 41
Büttiker’s Theory
Conductivity for a single free particle
Hence,
✓ ◆
✓
E
0
E=
)J=
0
e ⌫ E/
0
◆
Comparing to the definition of the conductivity tensor, we have
xx
Rhine Samajdar (Harvard University)
=0
xy
=
e⌫
0
The Integer Quantum Hall E↵ect
December 14, 2016
13 / 41
Büttiker’s Theory
Conductivity for a single free particle
Hence,
✓ ◆
✓
E
0
E=
)J=
0
e ⌫ E/
0
◆
Comparing to the definition of the conductivity tensor, we have
xx
=0
xy
=
e⌫
0
This is exactly the conductivity seen on the quantum Hall plateaus!
Rhine Samajdar (Harvard University)
The Integer Quantum Hall E↵ect
December 14, 2016
13 / 41
Büttiker’s Theory
Edge modes
Edge Modes
Figure: Classical picture (skipping
orbits).
Rhine Samajdar (Harvard University)
The Integer Quantum Hall E↵ect
December 14, 2016
14 / 41
Büttiker’s Theory
Edge modes
Edge Modes
Particles move in one direction on one
side of the sample, and in the other
direction on the other side i.e. the
particles have opposite chirality on the
two sides.
Figure: Classical picture (skipping
orbits).
Rhine Samajdar (Harvard University)
The Integer Quantum Hall E↵ect
December 14, 2016
14 / 41
Büttiker’s Theory
Edge modes
Edge Modes
Figure: E↵ective potential.
Rhine Samajdar (Harvard University)
The Integer Quantum Hall E↵ect
December 14, 2016
15 / 41
Büttiker’s Theory
Edge modes
Edge Modes
H=
1
p2 + (py + e B x)2 +V (x)
2m x
If the potential is smooth over
distance scales lB ,
V (x) ⇡ V (X) +
@V
(x
@x
X) + . . .
Figure: E↵ective potential.
Rhine Samajdar (Harvard University)
The Integer Quantum Hall E↵ect
December 14, 2016
15 / 41
Büttiker’s Theory
Edge modes
Edge Modes
H=
1
p2 + (py + e B x)2 +V (x)
2m x
If the potential is smooth over
distance scales lB ,
V (x) ⇡ V (X) +
Figure: E↵ective potential.
Rhine Samajdar (Harvard University)
@V
(x
@x
X) + . . .
Drift velocity along ŷ : vy =
The Integer Quantum Hall E↵ect
December 14, 2016
1 @V
eB @x
15 / 41
Büttiker’s Theory
Edge modes
Filling the edge states...
Figure: The bulk is an insulator.
Rhine Samajdar (Harvard University)
The Integer Quantum Hall E↵ect
December 14, 2016
16 / 41
Büttiker’s Theory
Edge modes
Filling the edge states...
Figure: Introduce a (chemical) potential
di↵erence µ.
Rhine Samajdar (Harvard University)
The Integer Quantum Hall E↵ect
December 14, 2016
16 / 41
Büttiker’s Theory
Edge modes
Filling the edge states...
Z
dk
vy (k)
2⇡
Z
e
1 @V
=
dx
2
eB @x
2⇡ lB
e
=
µ
h
Iy =
Rhine Samajdar (Harvard University)
The Integer Quantum Hall E↵ect
e
December 14, 2016
16 / 41
Büttiker’s Theory
Edge modes
Filling the edge states...
Z
dk
vy (k)
2⇡
Z
e
1 @V
=
dx
2
eB @x
2⇡ lB
e
=
µ
h
Iy =
e
The Hall voltage is e VH = µ,
giving us the Hall conductivity
xy
=
Iy
e2
= .
VH
h
[Büttiker, PRB 38, 9375 (1988).]
Rhine Samajdar (Harvard University)
The Integer Quantum Hall E↵ect
December 14, 2016
16 / 41
Büttiker’s Theory
Edge modes
Filling the edge states...
The e↵ective potential could be tilted
by an electric field. . .
Rhine Samajdar (Harvard University)
The Integer Quantum Hall E↵ect
December 14, 2016
17 / 41
Büttiker’s Theory
Edge modes
Filling the edge states...
. . . or simply be random—the Hall
conductivity remains quantized.
Rhine Samajdar (Harvard University)
The Integer Quantum Hall E↵ect
December 14, 2016
17 / 41
Büttiker’s Theory
Edge modes
Filling the edge states...
With n filled Landau levels, there are
. . . or simply be random—the Hall n chiral modes on each edge (as long
conductivity remains quantized.
as EF lies between levels).
Rhine Samajdar (Harvard University)
The Integer Quantum Hall E↵ect
December 14, 2016
17 / 41
Büttiker’s Theory
Edge modes
Robustness of the Hall State
The calculations above show that if an integer number of Landau levels
are filled, then the longitudinal and Hall resistivities are those observed on
the plateaus.
Rhine Samajdar (Harvard University)
The Integer Quantum Hall E↵ect
December 14, 2016
18 / 41
Büttiker’s Theory
Edge modes
Robustness of the Hall State
The calculations above show that if an integer number of Landau levels
are filled, then the longitudinal and Hall resistivities are those observed on
the plateaus.
Questions:
Why do these plateaus exist in the first place?
Rhine Samajdar (Harvard University)
The Integer Quantum Hall E↵ect
December 14, 2016
18 / 41
Büttiker’s Theory
Edge modes
Robustness of the Hall State
The calculations above show that if an integer number of Landau levels
are filled, then the longitudinal and Hall resistivities are those observed on
the plateaus.
Questions:
Why do these plateaus exist in the first place?
Why are there sharp jumps between di↵erent plateaus?
Rhine Samajdar (Harvard University)
The Integer Quantum Hall E↵ect
December 14, 2016
18 / 41
Büttiker’s Theory
Disorder and Localization
The role of disorder
Assume V ⌧ ~!B and |rV | ⌧
~ !B
lB
(weak disorder).
Figure: Density of states without [left panel] and with disorder [right panel].
Rhine Samajdar (Harvard University)
The Integer Quantum Hall E↵ect
December 14, 2016
19 / 41
Büttiker’s Theory
Disorder and Localization
Localization: Semiclassical argument
Disorder turns many of the quantum states from extended to localized.
Center of the cyclotron orbit:
X=x
⇡y
⇡x
; Y =y+
m !b
m !b
Time evolution:
@V
@Y
2 @V
ilB
@X
2
i~Ẋ = [X, H + V ] = ilB
i~Ẏ = [Y, H + V ] =
Thus, the center of mass drifts in a direction (Ẋ, Ẏ ) which is perpendicular
to rV i.e., the motion is along equipotentials.
Rhine Samajdar (Harvard University)
The Integer Quantum Hall E↵ect
December 14, 2016
20 / 41
Büttiker’s Theory
Disorder and Localization
Localization due to disorder
Figure: The localization of states due to disorder [left panel] and the resultant
density of states [right panel].
Rhine Samajdar (Harvard University)
The Integer Quantum Hall E↵ect
December 14, 2016
21 / 41
Büttiker’s Theory
Disorder and Localization
Summary: Presence of plateaus
Equipotentials which stretch from one side of a sample to another are
relatively rare but exist on the edge of the sample.
Rhine Samajdar (Harvard University)
The Integer Quantum Hall E↵ect
December 14, 2016
22 / 41
Büttiker’s Theory
Disorder and Localization
Summary: Presence of plateaus
Equipotentials which stretch from one side of a sample to another are
relatively rare but exist on the edge of the sample.
The states at the far edge of a band (either of high or low energy) are
localized. Only states close to the center of the band are extended.
Rhine Samajdar (Harvard University)
The Integer Quantum Hall E↵ect
December 14, 2016
22 / 41
Büttiker’s Theory
Disorder and Localization
Summary: Presence of plateaus
Equipotentials which stretch from one side of a sample to another are
relatively rare but exist on the edge of the sample.
The states at the far edge of a band (either of high or low energy) are
localized. Only states close to the center of the band are extended.
Since the localized states cannot contribute to the current, populating these states does not change the conductivity—this explains the
plateaus that are observed.
Rhine Samajdar (Harvard University)
The Integer Quantum Hall E↵ect
December 14, 2016
22 / 41
Laughlin’s Gedankenexperiment
Table of Contents
1
Introduction
The Classical Hall E↵ect
Integer Quantum Hall E↵ect
2
Landau Levels
Review
Conductivity: A quick
calculation
3
5
6
Büttiker’s Theory
Edge modes
Disorder and Localization
Rhine Samajdar (Harvard University)
4
7
Laughlin’s Gedankenexperiment
Gauge Invariance
Spectral Flow
Topology
Linear Response
Quantized Hall Conductivity
Particles on a Lattice
The TKNN formula
Peierls substitution
Chern Numbers
References
The Integer Quantum Hall E↵ect
December 14, 2016
23 / 41
Laughlin’s Gedankenexperiment
Gedankenexperiment geometry
Figure: Gedankenexperiment considered by Laughlin [left panel] and the Corbino
ring geometry [right] topologically equivalent to it [Laughlin, PRB 23, 5632
(1981)].
Rhine Samajdar (Harvard University)
The Integer Quantum Hall E↵ect
December 14, 2016
24 / 41
Laughlin’s Gedankenexperiment
Gauge Invariance
The role of Gauge Invariance
Corbino ring: In addition to the background magnetic field B which penetrates the sample, we can thread an additional flux through the center of
the ring.
Rhine Samajdar (Harvard University)
The Integer Quantum Hall E↵ect
December 14, 2016
25 / 41
Laughlin’s Gedankenexperiment
Gauge Invariance
The role of Gauge Invariance
Corbino ring: In addition to the background magnetic field B which penetrates the sample, we can thread an additional flux through the center of
the ring.
Suppose we increase
EMF E =
0 /T.
Rhine Samajdar (Harvard University)
slowly from 0 to
The Integer Quantum Hall E↵ect
0
= h/e. This induces an
December 14, 2016
25 / 41
Laughlin’s Gedankenexperiment
Gauge Invariance
The role of Gauge Invariance
Corbino ring: In addition to the background magnetic field B which penetrates the sample, we can thread an additional flux through the center of
the ring.
Suppose we increase
EMF E =
0 /T.
slowly from 0 to
0
= h/e. This induces an
Assume that n electrons are transferred from the inner circle to the
outer circle in this time. This would result in a radial current Ir =
n e/T .
Rhine Samajdar (Harvard University)
The Integer Quantum Hall E↵ect
December 14, 2016
25 / 41
Laughlin’s Gedankenexperiment
Gauge Invariance
The role of Gauge Invariance
Corbino ring: In addition to the background magnetic field B which penetrates the sample, we can thread an additional flux through the center of
the ring.
Suppose we increase
EMF E =
0 /T.
slowly from 0 to
0
= h/e. This induces an
Assume that n electrons are transferred from the inner circle to the
outer circle in this time. This would result in a radial current Ir =
n e/T .
Then,
⇢xy =
Rhine Samajdar (Harvard University)
E
h 1
= 2 .
Ir
e n
The Integer Quantum Hall E↵ect
December 14, 2016
25 / 41
Laughlin’s Gedankenexperiment
Gauge Invariance
The role of Gauge Invariance
Corbino ring: In addition to the background magnetic field B which penetrates the sample, we can thread an additional flux through the center of
the ring.
Suppose we increase
EMF E =
0 /T.
slowly from 0 to
0
= h/e. This induces an
Assume that n electrons are transferred from the inner circle to the
outer circle in this time. This would result in a radial current Ir =
n e/T .
Then,
⇢xy =
E
h 1
= 2 .
Ir
e n
Our task, therefore, is to argue that n electrons are indeed transferred across
the ring.
Rhine Samajdar (Harvard University)
The Integer Quantum Hall E↵ect
December 14, 2016
25 / 41
Laughlin’s Gedankenexperiment
Spectral Flow
Spectral Flow in Landau Levels
Using symmetric gauge, the wavefunctions in the lowest Landau level
are
2
m
|z|2 /4lB
.
m ⇠z e
q
2.
The mth wavefunction is strongly peaked at a radius r ⇡ 2mlB
Rhine Samajdar (Harvard University)
The Integer Quantum Hall E↵ect
December 14, 2016
26 / 41
Laughlin’s Gedankenexperiment
Spectral Flow
Spectral Flow in Landau Levels
Using symmetric gauge, the wavefunctions in the lowest Landau level
are
2
m
|z|2 /4lB
.
m ⇠z e
q
2.
The mth wavefunction is strongly peaked at a radius r ⇡ 2mlB
If we increase the flux from
m(
= 0) !
= 0 to
=
the wavefunctions shift as
= m+1 ( = 0).
q
2.
Each state moves outwards, to radius r = 2(m + 1)lB
Rhine Samajdar (Harvard University)
m(
0,
0)
The Integer Quantum Hall E↵ect
December 14, 2016
26 / 41
Laughlin’s Gedankenexperiment
Spectral Flow
Spectral Flow in Landau Levels
Using symmetric gauge, the wavefunctions in the lowest Landau level
are
2
m
|z|2 /4lB
.
m ⇠z e
q
2.
The mth wavefunction is strongly peaked at a radius r ⇡ 2mlB
If we increase the flux from
m(
= 0) !
= 0 to
m(
=
0,
the wavefunctions shift as
0)
= m+1 ( = 0).
q
2.
Each state moves outwards, to radius r = 2(m + 1)lB
Thus if all states in the Landau level are filled, a single electron is
transferred from the inner ring to the outer ring as the flux is increased.
Rhine Samajdar (Harvard University)
The Integer Quantum Hall E↵ect
December 14, 2016
26 / 41
Laughlin’s Gedankenexperiment
Spectral Flow
Spectral Flow in the Presence of Disorder
1
H =
2m

✓
◆ ✓
@
@
~
r
+
r @r @r
21
Rhine Samajdar (Harvard University)
~ @
eBr
e
i
+
+
r@
2
2⇡r
The Integer Quantum Hall E↵ect
◆2
+ V (r, )
December 14, 2016
27 / 41
Laughlin’s Gedankenexperiment
Spectral Flow
Spectral Flow in the Presence of Disorder
1
H =
2m

✓
◆ ✓
@
@
~
r
+
r @r @r
21
~ @
eBr
e
i
+
+
r@
2
2⇡r
◆2
+ V (r, )
We attempt to undo the flux by a gauge transformation
✓
◆
e
(r, ) ! exp
i
(r, )
h
Always possible for localized states; possible for extended states only when
is an integer multiple of 0 .
Rhine Samajdar (Harvard University)
The Integer Quantum Hall E↵ect
December 14, 2016
27 / 41
Laughlin’s Gedankenexperiment
Spectral Flow
Spectral Flow in the Presence of Disorder
1
H =
2m

✓
◆ ✓
@
@
~
r
+
r @r @r
21
~ @
eBr
e
i
+
+
r@
2
2⇡r
◆2
+ V (r, )
We attempt to undo the flux by a gauge transformation
✓
◆
e
(r, ) ! exp
i
(r, )
h
Always possible for localized states; possible for extended states only when
is an integer multiple of 0 .
Conclusion
Only the extended states undergo spectral flow; these alone must map
onto themselves. The localized states don’t change as increases.
Rhine Samajdar (Harvard University)
The Integer Quantum Hall E↵ect
December 14, 2016
27 / 41
Topology
Table of Contents
1
Introduction
The Classical Hall E↵ect
Integer Quantum Hall E↵ect
2
Landau Levels
Review
Conductivity: A quick
calculation
3
5
6
Büttiker’s Theory
Edge modes
Disorder and Localization
Rhine Samajdar (Harvard University)
4
7
Laughlin’s Gedankenexperiment
Gauge Invariance
Spectral Flow
Topology
Linear Response
Quantized Hall Conductivity
Particles on a Lattice
The TKNN formula
Peierls substitution
Chern Numbers
References
The Integer Quantum Hall E↵ect
December 14, 2016
28 / 41
Topology
Linear Response
Linear Response
xy
Z 1
1
dt ei!t h0 |[Jy (0), Jx (t)]| 0i
~! 0

i X h0 |Jy | ni hn |Jx | 0i h0 |Jx | ni hn |Jy | 0i
=
!
~! + En E0
~! + E0 En
=
n6=0
Rhine Samajdar (Harvard University)
The Integer Quantum Hall E↵ect
December 14, 2016
29 / 41
Topology
Linear Response
Linear Response
xy
Z 1
1
dt ei!t h0 |[Jy (0), Jx (t)]| 0i
~! 0

i X h0 |Jy | ni hn |Jx | 0i h0 |Jx | ni hn |Jy | 0i
=
!
~! + En E0
~! + E0 En
=
n6=0
Taking the DC (! ! 0) limit, we get
Kubo formula
xy
= i~
X h0 |Jy | ni hn |Jx | 0i
(En
n6=0
Rhine Samajdar (Harvard University)
h0 |Jx | ni hn |Jy | 0i
E0 ) 2
The Integer Quantum Hall E↵ect
December 14, 2016
29 / 41
Topology
Linear Response
The role of Topology
Consider the Hall system on a spatial torus T2 . We thread a uniform
magnetic field B through the torus.
Rhine Samajdar (Harvard University)
The Integer Quantum Hall E↵ect
December 14, 2016
30 / 41
Topology
Linear Response
The role of Topology
Consider the Hall system on a spatial torus T2 . We thread a uniform
magnetic field B through the torus.
Magnetic translation operators:
✓
◆
✓
d·p
T (d) = exp i
! exp
~
Rhine Samajdar (Harvard University)
ir + eA
id ·
~
The Integer Quantum Hall E↵ect
◆
December 14, 2016
30 / 41
Topology
Linear Response
The role of Topology
Consider the Hall system on a spatial torus T2 . We thread a uniform
magnetic field B through the torus.
Magnetic translation operators:
✓
◆
✓
d·p
T (d) = exp i
! exp
~
In Landau gauge, Ty Tx = e
i e B Lx Ly /~ T
ir + eA
id ·
~
x Ty ,
◆
so we must have
Dirac quantization condition
B Lx Ly =
Rhine Samajdar (Harvard University)
h
n; n 2 Z
e
The Integer Quantum Hall E↵ect
December 14, 2016
30 / 41
Topology
Linear Response
Adding Flux
Ax =
x
Lx
; Ay =
H=
y
Ly
+Bx
X Ji i
Li
i=x,y
Figure: Two fluxes are threaded through
the torus.
Rhine Samajdar (Harvard University)
The Integer Quantum Hall E↵ect
December 14, 2016
31 / 41
Topology
Linear Response
Adding Flux
Ax =
x
Lx
; Ay =
y
Ly
+Bx
X Ji i
Li
H=
i=x,y
To first order in perturbation theory,
|
0i
0
=|
0i
+
X hn| H| 0 i
|ni
En E0
n6=
0
Figure: Two fluxes are threaded through
the torus.
Rhine Samajdar (Harvard University)
The Integer Quantum Hall E↵ect
December 14, 2016
31 / 41
Topology
Linear Response
Adding Flux
Ax =
x
Lx
; Ay =
y
Ly
+Bx
X Ji i
Li
H=
i=x,y
To first order in perturbation theory,
|
Figure: Two fluxes are threaded through
the torus.
Rhine Samajdar (Harvard University)
0i
@
@
0
=|
0
i
The Integer Quantum Hall E↵ect
0i
=
+
X hn| H| 0 i
|ni
En E0
n6=
0
1 X hn|Ji | 0 i
|ni
Li
En E0
n6=
0
December 14, 2016
31 / 41
Topology
Quantized Hall Conductivity
Hall Conductivity and the Chern Number
Introduce dimensionless angular variables ✓i = 2⇡
Rhine Samajdar (Harvard University)
The Integer Quantum Hall E↵ect
i
0
, to parametrize T2 .
December 14, 2016
32 / 41
Topology
Quantized Hall Conductivity
Hall Conductivity and the Chern Number
Introduce dimensionless angular variables ✓i = 2⇡
⌧
@
Berry connection: Ai ( ) = i
0
0
@✓i
Rhine Samajdar (Harvard University)
The Integer Quantum Hall E↵ect
i
0
, to parametrize T2 .
December 14, 2016
32 / 41
Topology
Quantized Hall Conductivity
Hall Conductivity and the Chern Number
Introduce dimensionless angular variables ✓i = 2⇡
⌧
@
Berry connection: Ai ( ) = i
0
0
@✓i
Berry curvature:

⌧
@Ax @Ay
@
@ 0
Fxy =
= i
0
@✓y
@✓x
@✓y
@✓x
Rhine Samajdar (Harvard University)
The Integer Quantum Hall E↵ect
i
0
, to parametrize T2 .
@
@✓x
⌧
0
@ 0
@✓y
December 14, 2016
32 / 41
Topology
Quantized Hall Conductivity
Hall Conductivity and the Chern Number
Introduce dimensionless angular variables ✓i = 2⇡
⌧
@
Berry connection: Ai ( ) = i
0
0
@✓i
Berry curvature:

⌧
@Ax @Ay
@
@ 0
Fxy =
= i
0
@✓y
@✓x
@✓y
@✓x
xy
= i~

@
@
y
Rhine Samajdar (Harvard University)
⌧
0
@
@
@
0
x
@
x
⌧
0
The Integer Quantum Hall E↵ect
@
@
i
0
, to parametrize T2 .
@
@✓x
0
y
⌧
=
0
@ 0
@✓y
e2
Fxy
~
December 14, 2016
32 / 41
Topology
Quantized Hall Conductivity
Hall Conductivity and the Chern Number
Introduce dimensionless angular variables ✓i = 2⇡
⌧
@
Berry connection: Ai ( ) = i
0
0
@✓i
Berry curvature:

⌧
@Ax @Ay
@
@ 0
Fxy =
= i
0
@✓y
@✓x
@✓y
@✓x
xy
= i~

@
@
y
⌧
0
@
@
x
Averaging over all fluxes,
Z
e2
d2 ✓
Fxy =
xy =
~ T2 (2⇡)2
Rhine Samajdar (Harvard University)
@
0
@
x
⌧
0
@
@
i
0
@
@✓x
0
⌧
=
y
e2
1
C; C =
h
2⇡
The Integer Quantum Hall E↵ect
, to parametrize T2 .
Z
T2
0
@ 0
@✓y
e2
Fxy
~
d2 ✓ Fxy
December 14, 2016
32 / 41
Topology
Quantized Hall Conductivity
Hall Conductivity and the Chern Number
Introduce dimensionless angular variables ✓i = 2⇡
⌧
@
Berry connection: Ai ( ) = i
0
0
@✓i
Berry curvature:

⌧
@Ax @Ay
@
@ 0
Fxy =
= i
0
@✓y
@✓x
@✓y
@✓x
xy
= i~

@
@
y
⌧
0
@
@
x
Averaging over all fluxes,
Z
e2
d2 ✓
Fxy =
xy =
~ T2 (2⇡)2
Rhine Samajdar (Harvard University)
@
0
@
x
⌧
0
@
@
i
0
@
@✓x
0
⌧
=
y
e2
1
C; C =
h
2⇡
The Integer Quantum Hall E↵ect
, to parametrize T2 .
Z
T2
0
@ 0
@✓y
e2
Fxy
~
d2 ✓ Fxy 2 Z
December 14, 2016
32 / 41
Particles on a Lattice
Table of Contents
1
Introduction
The Classical Hall E↵ect
Integer Quantum Hall E↵ect
2
Landau Levels
Review
Conductivity: A quick
calculation
3
5
6
Büttiker’s Theory
Edge modes
Disorder and Localization
Rhine Samajdar (Harvard University)
4
7
Laughlin’s Gedankenexperiment
Gauge Invariance
Spectral Flow
Topology
Linear Response
Quantized Hall Conductivity
Particles on a Lattice
The TKNN formula
Peierls substitution
Chern Numbers
References
The Integer Quantum Hall E↵ect
December 14, 2016
33 / 41
Particles on a Lattice
The TKNN formula
TKNN invariants
Lattice momenta take values in the Brillouin zone:
⇡
⇡
< kx  ;
a
a
⇡
⇡
< ky 
b
b
The wavefunctions in a given band can be written in Bloch form as
k (x)
= ei k·x uk (x)
with uk (x) periodic on a unit cell.
Rhine Samajdar (Harvard University)
The Integer Quantum Hall E↵ect
December 14, 2016
34 / 41
Particles on a Lattice
The TKNN formula
TKNN invariants
Lattice momenta take values in the Brillouin zone:
⇡
⇡
< kx  ;
a
a
⇡
⇡
< ky 
b
b
The wavefunctions in a given band can be written in Bloch form as
k (x)
= ei k·x uk (x)
with uk (x) periodic on a unit cell.
Assumptions:
The single particle spectrum decomposes into bands.
Rhine Samajdar (Harvard University)
The Integer Quantum Hall E↵ect
December 14, 2016
34 / 41
Particles on a Lattice
The TKNN formula
TKNN invariants
Lattice momenta take values in the Brillouin zone:
⇡
⇡
< kx  ;
a
a
⇡
⇡
< ky 
b
b
The wavefunctions in a given band can be written in Bloch form as
k (x)
= ei k·x uk (x)
with uk (x) periodic on a unit cell.
Assumptions:
The single particle spectrum decomposes into bands.
The electrons are non-interacting.
Rhine Samajdar (Harvard University)
The Integer Quantum Hall E↵ect
December 14, 2016
34 / 41
Particles on a Lattice
The TKNN formula
TKNN invariants
Lattice momenta take values in the Brillouin zone:
⇡
⇡
< kx  ;
a
a
⇡
⇡
< ky 
b
b
The wavefunctions in a given band can be written in Bloch form as
k (x)
= ei k·x uk (x)
with uk (x) periodic on a unit cell.
Assumptions:
The single particle spectrum decomposes into bands.
The electrons are non-interacting.
There is a gap between bands and the Fermi energy EF lies in one of
these gaps.
Rhine Samajdar (Harvard University)
The Integer Quantum Hall E↵ect
December 14, 2016
34 / 41
Particles on a Lattice
The TKNN formula
TKNN invariants
Berry connection: Ai (k) =
Rhine Samajdar (Harvard University)
⌧
i uk
@
uk
@k i
The Integer Quantum Hall E↵ect
December 14, 2016
35 / 41
Particles on a Lattice
The TKNN formula
TKNN invariants
Berry connection: Ai (k) =
Berry curvature:
@Ax @Ay
Fxy =
=
@k y
@k x
Rhine Samajdar (Harvard University)
i
⌧
⌧
i uk
@
uk
@k i
@u @u
@k y @k x
+i
The Integer Quantum Hall E↵ect
⌧
@u @u
@k x @k y
December 14, 2016
35 / 41
Particles on a Lattice
The TKNN formula
TKNN invariants
⌧
i uk
Berry connection: Ai (k) =
Berry curvature:
@Ax @Ay
Fxy =
=
@k y
@k x
Chern number: C =
Rhine Samajdar (Harvard University)
i
1
2⇡
⌧
Z
@
uk
@k i
@u @u
@k y @k x
T2
+i
⌧
@u @u
@k x @k y
d2 k Fxy 2 Z
The Integer Quantum Hall E↵ect
December 14, 2016
35 / 41
Particles on a Lattice
The TKNN formula
TKNN invariants
⌧
i uk
Berry connection: Ai (k) =
Berry curvature:
@Ax @Ay
Fxy =
=
@k y
@k x
Chern number: C =
i
1
2⇡
⌧
Z
@
uk
@k i
@u @u
@k y @k x
T2
+i
⌧
@u @u
@k x @k y
d2 k Fxy 2 Z
More generally, assigning a Chern number to each band ↵, we get:
TKNN formula
xy
=
e2 X
C↵ (Topological invariant)
h ↵
[Thouless, Kohomoto, Nightingale, and den Nijs, PRL 49, 405 (1982).]
Rhine Samajdar (Harvard University)
The Integer Quantum Hall E↵ect
December 14, 2016
35 / 41
Particles on a Lattice
Peierls substitution
Adding a magnetic field
Tight-binding model: H =
Rhine Samajdar (Harvard University)
t
P P
x
j=1,2 |xihx
The Integer Quantum Hall E↵ect
+ ej |+ h.c.
December 14, 2016
36 / 41
Particles on a Lattice
Peierls substitution
Adding a magnetic field
Tight-binding model: H =
Rhine Samajdar (Harvard University)
t
P P
x
j=1,2 |xie
The Integer Quantum Hall E↵ect
ie a Aj (x)/~ hx
+ ej |+ h.c.
December 14, 2016
36 / 41
Particles on a Lattice
Peierls substitution
Adding a magnetic field
Tight-binding model: H =
t
P P
x
j=1,2 |xie
ie a Aj (x)/~ hx
+ ej |+ h.c.
Figure: Consider a particle which moves
anti-clockwise around this plaquette.
Rhine Samajdar (Harvard University)
The Integer Quantum Hall E↵ect
December 14, 2016
36 / 41
Particles on a Lattice
Peierls substitution
Adding a magnetic field
Tight-binding model: H =
t
P P
x
j=1,2 |xie
ie a Aj (x)/~ hx
+ ej |+ h.c.
To leading order in t, it will pick up a
phase e i , where
eA
A1 (x) + A2 (x + e1 )
~
A1 (x + e2 ) A2 (x)
✓
◆
e A2 @A2 @A1
eA2 B
⇡
=
~
@x1
@x2
~
=
Figure: Consider a particle which moves
anti-clockwise around this plaquette.
Rhine Samajdar (Harvard University)
The Integer Quantum Hall E↵ect
December 14, 2016
36 / 41
Particles on a Lattice
Peierls substitution
Adding a magnetic field
Tight-binding model: H =
t
P P
x
j=1,2 |xie
ie a Aj (x)/~ hx
+ ej |+ h.c.
To leading order in t, it will pick up a
phase e i , where
eA
A1 (x) + A2 (x + e1 )
~
A1 (x + e2 ) A2 (x)
✓
◆
e A2 @A2 @A1
eA2 B
⇡
=
~
@x1
@x2
~
=
Figure: Consider a particle which moves
anti-clockwise around this plaquette.
Rhine Samajdar (Harvard University)
Aharonov-Bohm phase!
The Integer Quantum Hall E↵ect
December 14, 2016
36 / 41
Particles on a Lattice
Peierls substitution
Magnetic Brillouin zone
Construct operators: T̃j =
P
x |xie
ie a Ãj (x)/~ hx
+ ej |.
New gauge field: @k Ãj = @j Ak . In Landau gauge, Ã1 = B x2 , Ã2 = 0.
Rhine Samajdar (Harvard University)
The Integer Quantum Hall E↵ect
December 14, 2016
37 / 41
Particles on a Lattice
Peierls substitution
Magnetic Brillouin zone
Construct operators: T̃j =
P
x |xie
ie a Ãj (x)/~ hx
+ ej |.
New gauge field: @k Ãj = @j Ak . In Landau gauge, Ã1 = B x2 , Ã2 = 0.
We can build commuting operators by [T̃1n1 , T̃2n2 ], when
p
q
Rhine Samajdar (Harvard University)
December 14, 2016
The Integer Quantum Hall E↵ect
n1 n2 2 Z.
37 / 41
Particles on a Lattice
Peierls substitution
Magnetic Brillouin zone
Construct operators: T̃j =
P
x |xie
ie a Ãj (x)/~ hx
+ ej |.
New gauge field: @k Ãj = @j Ak . In Landau gauge, Ã1 = B x2 , Ã2 = 0.
We can build commuting operators by [T̃1n1 , T̃2n2 ], when
p
q
n1 n2 2 Z.
Bloch-like eigenstates: H|ki = E(k) |ki with T1q |ki = eiq k1 a |ki.
Rhine Samajdar (Harvard University)
The Integer Quantum Hall E↵ect
December 14, 2016
37 / 41
Particles on a Lattice
Peierls substitution
Magnetic Brillouin zone
Construct operators: T̃j =
P
x |xie
ie a Ãj (x)/~ hx
+ ej |.
New gauge field: @k Ãj = @j Ak . In Landau gauge, Ã1 = B x2 , Ã2 = 0.
We can build commuting operators by [T̃1n1 , T̃2n2 ], when
p
q
n1 n2 2 Z.
Bloch-like eigenstates: H|ki = E(k) |ki with T1q |ki = eiq k1 a |ki.
Magnetic Brillouin zone
⇡
⇡
< k1 
;
qa
qa
Rhine Samajdar (Harvard University)
⇡
⇡
< k2 
a
a
The Integer Quantum Hall E↵ect
December 14, 2016
37 / 41
Particles on a Lattice
Peierls substitution
Energy Spectrum
The spectrum decomposes into
q bands, each with a di↵erent
range of energies.
Rhine Samajdar (Harvard University)
The Integer Quantum Hall E↵ect
December 14, 2016
38 / 41
Particles on a Lattice
Peierls substitution
Energy Spectrum
The spectrum decomposes into
q bands, each with a di↵erent
range of energies.
Any energy eigenvalue in a given
band is q-fold degenerate as |ki
has the same energy as T̃1 |ki ⇠
|k1 , k2 + 2⇡p/qai.
Rhine Samajdar (Harvard University)
The Integer Quantum Hall E↵ect
December 14, 2016
38 / 41
Particles on a Lattice
Peierls substitution
Energy Spectrum
The spectrum decomposes into
q bands, each with a di↵erent
range of energies.
Any energy eigenvalue in a given
band is q-fold degenerate as |ki
has the same energy as T̃1 |ki ⇠
|k1 , k2 + 2⇡p/qai.
Rhine Samajdar (Harvard University)
Figure: Hofstadter butterfly.[Hofstadter,
PRB 14, 2239 (1976).]
The Integer Quantum Hall E↵ect
December 14, 2016
38 / 41
Particles on a Lattice
Chern Numbers
TKNN Invariants on a Lattice
We can compute the Hall conductivity only for rational fluxes
for which there exists a magnetic Brillouin zone.
Rhine Samajdar (Harvard University)
The Integer Quantum Hall E↵ect
=p
December 14, 2016
0 /q
39 / 41
Particles on a Lattice
Chern Numbers
TKNN Invariants on a Lattice
We can compute the Hall conductivity only for rational fluxes = p 0 /q
for which there exists a magnetic Brillouin zone.
First consider the rth of the q bands. Then, to compute the Chern
number, we have to solve the linear Diophantine equation
q
r = q sr + p tr ; |tr |  .
2
Rhine Samajdar (Harvard University)
The Integer Quantum Hall E↵ect
December 14, 2016
39 / 41
Particles on a Lattice
Chern Numbers
TKNN Invariants on a Lattice
We can compute the Hall conductivity only for rational fluxes = p 0 /q
for which there exists a magnetic Brillouin zone.
First consider the rth of the q bands. Then, to compute the Chern
number, we have to solve the linear Diophantine equation
q
r = q sr + p tr ; |tr |  .
2
The Chern number of the rth band is C = tr
Rhine Samajdar (Harvard University)
The Integer Quantum Hall E↵ect
tr
1,
with t0 ⌘ 0.
December 14, 2016
39 / 41
Particles on a Lattice
Chern Numbers
TKNN Invariants on a Lattice
We can compute the Hall conductivity only for rational fluxes = p 0 /q
for which there exists a magnetic Brillouin zone.
First consider the rth of the q bands. Then, to compute the Chern
number, we have to solve the linear Diophantine equation
q
r = q sr + p tr ; |tr |  .
2
The Chern number of the rth band is C = tr tr 1 , with t0 ⌘ 0.
e2
If Er < EF < Er+1 , then xy = tr [TKNN, PRL 49, 405 (1982)].
h
Rhine Samajdar (Harvard University)
The Integer Quantum Hall E↵ect
December 14, 2016
39 / 41
References
Table of Contents
1
Introduction
The Classical Hall E↵ect
Integer Quantum Hall E↵ect
2
Landau Levels
Review
Conductivity: A quick
calculation
3
5
6
Büttiker’s Theory
Edge modes
Disorder and Localization
Rhine Samajdar (Harvard University)
4
7
Laughlin’s Gedankenexperiment
Gauge Invariance
Spectral Flow
Topology
Linear Response
Quantized Hall Conductivity
Particles on a Lattice
The TKNN formula
Peierls substitution
Chern Numbers
References
The Integer Quantum Hall E↵ect
December 14, 2016
40 / 41
References
References and Acknowledgments
D. Tong, The Quantum Hall E↵ect, TIFR Infosys Lectures (2016).
D. Yoshioka, The Quantum Hall E↵ect, Springer-Verlag (2002).
Rhine Samajdar (Harvard University)
The Integer Quantum Hall E↵ect
December 14, 2016
41 / 41
References
References and Acknowledgments
D. Tong, The Quantum Hall E↵ect, TIFR Infosys Lectures (2016).
D. Yoshioka, The Quantum Hall E↵ect, Springer-Verlag (2002).
I acknowledge useful discussions with Aavishkar A. Patel.
Rhine Samajdar (Harvard University)
The Integer Quantum Hall E↵ect
December 14, 2016
41 / 41
References
References and Acknowledgments
D. Tong, The Quantum Hall E↵ect, TIFR Infosys Lectures (2016).
D. Yoshioka, The Quantum Hall E↵ect, Springer-Verlag (2002).
I acknowledge useful discussions with Aavishkar A. Patel.
Thank you!
Rhine Samajdar (Harvard University)
The Integer Quantum Hall E↵ect
December 14, 2016
41 / 41