Post on 04-Feb-2021
The 2016 Nobel prize in PhysicsD. Thouless and Topological Invariants
J. Avron
January 2017
Avron The 2016 Nobel prize in Physics: January 2017 1 / 25
David J. Thouless
D. Thouless, D. Haldane, M. KosteritzKosterlitz-Thouless transition; TKNN aka Chern numbers
Avron The 2016 Nobel prize in Physics: January 2017 2 / 25
TKNN
TKNN 1982cited 2874
M.V. Berry: Phase ... in adiabatic changes (1984) (cited 7200)B. Simon: Holonomy ... and Berry’s phase (1983)
Avron The 2016 Nobel prize in Physics: January 2017 3 / 25
TKNN
The Classical Hall effect:Ingenious errors 1879
I
V
B
1/B
I/V
T � B
Avron The 2016 Nobel prize in Physics: January 2017 4 / 25
TKNN
The Classical Hall effect:Ingenious errors 1879
I
V
B
1/B
I/V
T � B
Avron The 2016 Nobel prize in Physics: January 2017 4 / 25
TKNN
The Classical Hall effect:Ingenious errors 1879
I
V
B
1/B
I/V
T � B
Avron The 2016 Nobel prize in Physics: January 2017 4 / 25
The Quantum Hall effect
The Quantum Hall effectvon Klitzing (Nobel 1985)
Quantum unit of resistance
he2≈ 26 [K Ω]
1/B
I/V [e2 h−1]
1.000000000001
1.999999999999
3.000000000006
T � B
Avron The 2016 Nobel prize in Physics: January 2017 5 / 25
The Quantum Hall effect
The Quantum Hall effectvon Klitzing (Nobel 1985)
Quantum unit of resistance
he2≈ 26 [K Ω]
1/B
I/V [e2 h−1]
1.000000000001
1.999999999999
3.000000000006
T � B
Avron The 2016 Nobel prize in Physics: January 2017 5 / 25
The Quantum Hall effect
The Quantum Hall effectvon Klitzing (Nobel 1985)
Quantum unit of resistance
he2≈ 26 [K Ω]
1/B
I/V [e2 h−1]
1.000000000001
1.999999999999
3.000000000006
T � B
Avron The 2016 Nobel prize in Physics: January 2017 5 / 25
The Quantum Hall effect
Fundamental vs natural standards
Second: Hyperfine transition of Cs133
All Cs133 atoms are the same
9,192,631,770 [Hz]
Natural but not fundamental
Ohm: QHE
I
V
Every QHS is different
1/B
e2/h
1
2
3
Artificial but fundamental
Avron The 2016 Nobel prize in Physics: January 2017 6 / 25
The Quantum Hall effect
Fundamental vs natural standards
Second: Hyperfine transition of Cs133
All Cs133 atoms are the same
9,192,631,770 [Hz]
Natural but not fundamental
Ohm: QHE
I
V
Every QHS is different
1/B
e2/h
1
2
3
Artificial but fundamental
Avron The 2016 Nobel prize in Physics: January 2017 6 / 25
The Quantum Hall effect
Fundamental vs natural standards
Second: Hyperfine transition of Cs133
All Cs133 atoms are the same
9,192,631,770 [Hz]
Natural but not fundamental
Ohm: QHE
I
V
Every QHS is different
1/B
e2/h
1
2
3
Artificial but fundamental
Avron The 2016 Nobel prize in Physics: January 2017 6 / 25
The Quantum Hall effect
Fundamental vs natural standards
Second: Hyperfine transition of Cs133
All Cs133 atoms are the same
9,192,631,770 [Hz]
Natural but not fundamental
Ohm: QHE
I
V
Every QHS is different
1/B
e2/h
1
2
3
Artificial but fundamental
Avron The 2016 Nobel prize in Physics: January 2017 6 / 25
The Quantum Hall effect
The Problem
1/B
I/V
Conductance:∂V 〈ψ |I|ψ〉
∣∣∣V =0︸ ︷︷ ︸
Derivative of expectation
What is the mechanism of quantization?Why integer multiple of h/e2?
Avron The 2016 Nobel prize in Physics: January 2017 7 / 25
The Quantum Hall effect
The Problem
1/B
I/V
Conductance:∂V 〈ψ |I|ψ〉
∣∣∣V =0︸ ︷︷ ︸
Derivative of expectation
What is the mechanism of quantization?Why integer multiple of h/e2?
Avron The 2016 Nobel prize in Physics: January 2017 7 / 25
The Quantum Hall effect
The Problem
1/B
I/V
Conductance:∂V 〈ψ |I|ψ〉
∣∣∣V =0︸ ︷︷ ︸
Derivative of expectation
What is the mechanism of quantization?Why integer multiple of h/e2?
Avron The 2016 Nobel prize in Physics: January 2017 7 / 25
The Quantum Hall effect
Heisenberg, Dirac and TKNNThree types of quantizations in QM
Quantized observables
Spect(Lz) ⊆~2Z︸ ︷︷ ︸
quantized observables
〈ψ|Lz |ψ〉 ⊆~2R︸ ︷︷ ︸
Expectations–Not quantized
Quantized parameters
qeqm ∈~c2
Z︸ ︷︷ ︸qm:charge of monopole
Quantized expectations
∂V 〈ψ |I|ψ〉∣∣∣V =0∈ e
2
hZ
B
Avron The 2016 Nobel prize in Physics: January 2017 8 / 25
The Quantum Hall effect
Heisenberg, Dirac and TKNNThree types of quantizations in QM
Quantized observables
Spect(Lz) ⊆~2Z︸ ︷︷ ︸
quantized observables
〈ψ|Lz |ψ〉 ⊆~2R︸ ︷︷ ︸
Expectations–Not quantized
Quantized parameters
qeqm ∈~c2
Z︸ ︷︷ ︸qm:charge of monopole
Quantized expectations
∂V 〈ψ |I|ψ〉∣∣∣V =0∈ e
2
hZ
B
Avron The 2016 Nobel prize in Physics: January 2017 8 / 25
The Quantum Hall effect
Heisenberg, Dirac and TKNNThree types of quantizations in QM
Quantized observables
Spect(Lz) ⊆~2Z︸ ︷︷ ︸
quantized observables
〈ψ|Lz |ψ〉 ⊆~2R︸ ︷︷ ︸
Expectations–Not quantized
Quantized parameters
qeqm ∈~c2
Z︸ ︷︷ ︸qm:charge of monopole
Quantized expectations
∂V 〈ψ |I|ψ〉∣∣∣V =0∈ e
2
hZ
B
Avron The 2016 Nobel prize in Physics: January 2017 8 / 25
The Quantum Hall effect
Heisenberg, Dirac and TKNNThree types of quantizations in QM
Quantized observables
Spect(Lz) ⊆~2Z︸ ︷︷ ︸
quantized observables
〈ψ|Lz |ψ〉 ⊆~2R︸ ︷︷ ︸
Expectations–Not quantized
Quantized parameters
qeqm ∈~c2
Z︸ ︷︷ ︸qm:charge of monopole
Quantized expectations
∂V 〈ψ |I|ψ〉∣∣∣V =0∈ e
2
hZ
B
Avron The 2016 Nobel prize in Physics: January 2017 8 / 25
The Quantum Hall effect
Heisenberg, Dirac and TKNNThree types of quantizations in QM
Quantized observables
Spect(Lz) ⊆~2Z︸ ︷︷ ︸
quantized observables
〈ψ|Lz |ψ〉 ⊆~2R︸ ︷︷ ︸
Expectations–Not quantized
Quantized parameters
qeqm ∈~c2
Z︸ ︷︷ ︸qm:charge of monopole
Quantized expectations
∂V 〈ψ |I|ψ〉∣∣∣V =0∈ e
2
hZ
B
Avron The 2016 Nobel prize in Physics: January 2017 8 / 25
The Quantum Hall effect
Heisenberg, Dirac and TKNNThree types of quantizations in QM
Quantized observables
Spect(Lz) ⊆~2Z︸ ︷︷ ︸
quantized observables
〈ψ|Lz |ψ〉 ⊆~2R︸ ︷︷ ︸
Expectations–Not quantized
Quantized parameters
qeqm ∈~c2
Z︸ ︷︷ ︸qm:charge of monopole
Quantized expectations
∂V 〈ψ |I|ψ〉∣∣∣V =0∈ e
2
hZ
B
Avron The 2016 Nobel prize in Physics: January 2017 8 / 25
The Quantum Hall effect
Heisenberg, Dirac and TKNNThree types of quantizations in QM
Quantized observables
Spect(Lz) ⊆~2Z︸ ︷︷ ︸
quantized observables
〈ψ|Lz |ψ〉 ⊆~2R︸ ︷︷ ︸
Expectations–Not quantized
Quantized parameters
qeqm ∈~c2
Z︸ ︷︷ ︸qm:charge of monopole
Quantized expectations
∂V 〈ψ |I|ψ〉∣∣∣V =0∈ e
2
hZ
B
Avron The 2016 Nobel prize in Physics: January 2017 8 / 25
The Quantum Hall effect
Real scientists solve models. Wimps generalize M. BerryThe Hofstadter model
Φ
E
N
Magnetic flux
Hopping on lattice in a homogeneous magnetic field
H = E + N + h.c.
East and North translations(Eψ)(n,m) = e−2πiΦ mψ(n − 1,m),
(Nψ)(n,m) = ψ(n,m − 1)
Non-commutativeEN = e−2πiΦNE
Avron The 2016 Nobel prize in Physics: January 2017 9 / 25
The Quantum Hall effect
Real scientists solve models. Wimps generalize M. BerryThe Hofstadter model
Φ
E
N
Magnetic flux
Hopping on lattice in a homogeneous magnetic field
H = E + N + h.c.
East and North translations(Eψ)(n,m) = e−2πiΦ mψ(n − 1,m),
(Nψ)(n,m) = ψ(n,m − 1)
Non-commutativeEN = e−2πiΦNE
Avron The 2016 Nobel prize in Physics: January 2017 9 / 25
The Quantum Hall effect
Real scientists solve models. Wimps generalize M. BerryThe Hofstadter model
Φ
E
N
Magnetic flux
Hopping on lattice in a homogeneous magnetic field
H = E + N + h.c.
East and North translations(Eψ)(n,m) = e−2πiΦ mψ(n − 1,m),
(Nψ)(n,m) = ψ(n,m − 1)
Non-commutativeEN = e−2πiΦNE
Avron The 2016 Nobel prize in Physics: January 2017 9 / 25
The Quantum Hall effect
Real scientists solve models. Wimps generalize M. BerryThe Hofstadter model
Φ
E
N
Magnetic flux
Hopping on lattice in a homogeneous magnetic field
H = E + N + h.c.
East and North translations(Eψ)(n,m) = e−2πiΦ mψ(n − 1,m),
(Nψ)(n,m) = ψ(n,m − 1)
Non-commutativeEN = e−2πiΦNE
Avron The 2016 Nobel prize in Physics: January 2017 9 / 25
The Quantum Hall effect
The importance of familiesDoubly periodic matrices
Bloch decomposition Φ = pq :
H(k1, k2)︸ ︷︷ ︸q×q periodic matrix
= eik1 T1︸︷︷︸shift
+eik2 T2︸︷︷︸boost
+h.c.
Example Φ = 13 :
T1 =
0 1 00 0 11 0 0
︸ ︷︷ ︸
3×3
, T2 =
1 0 00 ω 00 0 ω2
, ω = e2πiΦ︸ ︷︷ ︸root of unity
Avron The 2016 Nobel prize in Physics: January 2017 10 / 25
The Quantum Hall effect
The importance of familiesDoubly periodic matrices
Bloch decomposition Φ = pq :
H(k1, k2)︸ ︷︷ ︸q×q periodic matrix
= eik1 T1︸︷︷︸shift
+eik2 T2︸︷︷︸boost
+h.c.
Example Φ = 13 :
T1 =
0 1 00 0 11 0 0
︸ ︷︷ ︸
3×3
, T2 =
1 0 00 ω 00 0 ω2
, ω = e2πiΦ︸ ︷︷ ︸root of unity
Avron The 2016 Nobel prize in Physics: January 2017 10 / 25
The Quantum Hall effect
Gauss Chern and PretzlesFiber bundles for pedestrians
12π
∫Curvature = 2(1− g) ∈ Z
Avron The 2016 Nobel prize in Physics: January 2017 11 / 25
The Quantum Hall effect
Chern numbersTKNN, Simon, ASS, Bellissard
Bloch Electrons
H(k) = H(k + 2π)
Controlled Hamiltonian
φ2φ1
H(φ) = H(φ+ 2π)
period
+1
-2
Avron The 2016 Nobel prize in Physics: January 2017 12 / 25
The Quantum Hall effect
Chern numbersTKNN, Simon, ASS, Bellissard
Bloch Electrons
H(k) = H(k + 2π)
Controlled Hamiltonian
φ2φ1
H(φ) = H(φ+ 2π)
period
+1
-2
Avron The 2016 Nobel prize in Physics: January 2017 12 / 25
The Quantum Hall effect
Chern numbersTKNN, Simon, ASS, Bellissard
Bloch Electrons
H(k) = H(k + 2π)
Controlled Hamiltonian
φ2φ1
H(φ) = H(φ+ 2π)
period
+1
-2
Avron The 2016 Nobel prize in Physics: January 2017 12 / 25
The Quantum Hall effect
TKNN discoveryQuantization of Hall conductance
Brillouin zone
EF
Fille
d+1
-1
Full band contributes an integer:
∂V 〈ψ |I|ψ〉∣∣∣V =0
=e2
hσ︸︷︷︸
Integer
Diophantine equation (TKNN,DAZ)
σp = 1 mod q
Avron The 2016 Nobel prize in Physics: January 2017 13 / 25
The Quantum Hall effect
TKNN discoveryQuantization of Hall conductance
Brillouin zone
EF
Fille
d+1
-1
Full band contributes an integer:
∂V 〈ψ |I|ψ〉∣∣∣V =0
=e2
hσ︸︷︷︸
Integer
Diophantine equation (TKNN,DAZ)
σp = 1 mod q
Avron The 2016 Nobel prize in Physics: January 2017 13 / 25
The Quantum Hall effect
TKNN discoveryQuantization of Hall conductance
Brillouin zone
EF
Fille
d+1
-1
Full band contributes an integer:
∂V 〈ψ |I|ψ〉∣∣∣V =0
=e2
hσ︸︷︷︸
Integer
Diophantine equation (TKNN,DAZ)
σp = 1 mod q
Avron The 2016 Nobel prize in Physics: January 2017 13 / 25
The Quantum Hall effect
TKNN discoveryQuantization of Hall conductance
Brillouin zone
EF
Fille
d+1
-1
Full band contributes an integer:
∂V 〈ψ |I|ψ〉∣∣∣V =0
=e2
hσ︸︷︷︸
Integer
Diophantine equation (TKNN,DAZ)
σp = 1 mod q
Avron The 2016 Nobel prize in Physics: January 2017 13 / 25
The Quantum Hall effect
TKNN discoveryQuantization of Hall conductance
Brillouin zone
EF
Fille
d+1
-1
Full band contributes an integer:
∂V 〈ψ |I|ψ〉∣∣∣V =0
=e2
hσ︸︷︷︸
Integer
Diophantine equation (TKNN,DAZ)
σp = 1 mod q
Avron The 2016 Nobel prize in Physics: January 2017 13 / 25
The Quantum Hall effect
Hofstadter butterflyTKNN integers everywhere
EF
Φ
Avron The 2016 Nobel prize in Physics: January 2017 14 / 25
The Quantum Hall effect
A geometric perspectiveWhat is Berry’s phase? (Berry, Simon)
Period
|ψ〉 eiβ |ψ〉
−π π
A = i〈ψ|∇kψ〉︸ ︷︷ ︸Berry’s gauge potential
Periodic projection
|ψ〉 〈ψ| (−π) = |ψ〉 〈ψ| (π)
Berry’s phase
|ψ〉 (π) = eiβ |ψ〉 (−π)
Berry’s gauge potential
β =
∮A(k) · dk
Berry’s curvature ∇× A
Avron The 2016 Nobel prize in Physics: January 2017 15 / 25
The Quantum Hall effect
A geometric perspectiveWhat is Berry’s phase? (Berry, Simon)
Period
|ψ〉 eiβ |ψ〉
−π π
A = i〈ψ|∇kψ〉︸ ︷︷ ︸Berry’s gauge potential
Periodic projection
|ψ〉 〈ψ| (−π) = |ψ〉 〈ψ| (π)
Berry’s phase
|ψ〉 (π) = eiβ |ψ〉 (−π)
Berry’s gauge potential
β =
∮A(k) · dk
Berry’s curvature ∇× A
Avron The 2016 Nobel prize in Physics: January 2017 15 / 25
The Quantum Hall effect
A geometric perspectiveWhat is Berry’s phase? (Berry, Simon)
Period
|ψ〉 eiβ |ψ〉
−π π
A = i〈ψ|∇kψ〉︸ ︷︷ ︸Berry’s gauge potential
Periodic projection
|ψ〉 〈ψ| (−π) = |ψ〉 〈ψ| (π)
Berry’s phase
|ψ〉 (π) = eiβ |ψ〉 (−π)
Berry’s gauge potential
β =
∮A(k) · dk
Berry’s curvature ∇× A
Avron The 2016 Nobel prize in Physics: January 2017 15 / 25
The Quantum Hall effect
A geometric perspectiveWhat is Berry’s phase? (Berry, Simon)
Period
|ψ〉 eiβ |ψ〉
−π π
A = i〈ψ|∇kψ〉︸ ︷︷ ︸Berry’s gauge potential
Periodic projection
|ψ〉 〈ψ| (−π) = |ψ〉 〈ψ| (π)
Berry’s phase
|ψ〉 (π) = eiβ |ψ〉 (−π)
Berry’s gauge potential
β =
∮A(k) · dk
Berry’s curvature ∇× A
Avron The 2016 Nobel prize in Physics: January 2017 15 / 25
The Quantum Hall effect
A geometric perspectiveWhat is Berry’s phase? (Berry, Simon)
Period
|ψ〉 eiβ |ψ〉
−π π
A = i〈ψ|∇kψ〉︸ ︷︷ ︸Berry’s gauge potential
Periodic projection
|ψ〉 〈ψ| (−π) = |ψ〉 〈ψ| (π)
Berry’s phase
|ψ〉 (π) = eiβ |ψ〉 (−π)
Berry’s gauge potential
β =
∮A(k) · dk
Berry’s curvature ∇× A
Avron The 2016 Nobel prize in Physics: January 2017 15 / 25
The Quantum Hall effect
A geometric perspectiveWhat is Berry’s phase? (Berry, Simon)
Period
|ψ〉 eiβ |ψ〉
−π π
A = i〈ψ|∇kψ〉︸ ︷︷ ︸Berry’s gauge potential
Periodic projection
|ψ〉 〈ψ| (−π) = |ψ〉 〈ψ| (π)
Berry’s phase
|ψ〉 (π) = eiβ |ψ〉 (−π)
Berry’s gauge potential
β =
∮A(k) · dk
Berry’s curvature ∇× A
Avron The 2016 Nobel prize in Physics: January 2017 15 / 25
The Quantum Hall effect
A geometric perspectiveWhat is Berry’s phase? (Berry, Simon)
Period
|ψ〉 eiβ |ψ〉
−π π
A = i〈ψ|∇kψ〉︸ ︷︷ ︸Berry’s gauge potential
Periodic projection
|ψ〉 〈ψ| (−π) = |ψ〉 〈ψ| (π)
Berry’s phase
|ψ〉 (π) = eiβ |ψ〉 (−π)
Berry’s gauge potential
β =
∮A(k) · dk
Berry’s curvature ∇× A
Avron The 2016 Nobel prize in Physics: January 2017 15 / 25
The Quantum Hall effect
Conductance= Adiabatic CurvatureSimon, A.-Seiler, Bellissard, Niu-Thouless
k1,2
EF
Fille
dTKNN
∂V 〈ψ |I|ψ〉∣∣∣V =0
=e2
h×∑
j∈filled
∫BZ
dk1dk2 ∇k × Aψj︸ ︷︷ ︸Berry’s curvature
Avron The 2016 Nobel prize in Physics: January 2017 16 / 25
The Quantum Hall effect
Conductance= Adiabatic CurvatureSimon, A.-Seiler, Bellissard, Niu-Thouless
k1,2
EF
Fille
dTKNN
∂V 〈ψ |I|ψ〉∣∣∣V =0
=e2
h×∑
j∈filled
∫BZ
dk1dk2 ∇k × Aψj︸ ︷︷ ︸Berry’s curvature
Avron The 2016 Nobel prize in Physics: January 2017 16 / 25
The Quantum Hall effect
Conductance= Adiabatic CurvatureSimon, A.-Seiler, Bellissard, Niu-Thouless
k1,2
EF
Fille
dTKNN
∂V 〈ψ |I|ψ〉∣∣∣V =0
=e2
h×∑
j∈filled
∫BZ
dk1dk2 ∇k × Aψj︸ ︷︷ ︸Berry’s curvature
Avron The 2016 Nobel prize in Physics: January 2017 16 / 25
The Quantum Hall effect
Conductance= Adiabatic CurvatureSimon, A.-Seiler, Bellissard, Niu-Thouless
k1,2
EF
Fille
dTKNN
∂V 〈ψ |I|ψ〉∣∣∣V =0
=e2
h×∑
j∈filled
∫BZ
dk1dk2 ∇k × Aψj︸ ︷︷ ︸Berry’s curvature
Avron The 2016 Nobel prize in Physics: January 2017 16 / 25
The Quantum Hall effect
Chern numbers and Berry’s phase
Stokes∫BZ∇k × A dk1 dk2 =
∮∂BZ
dk · ABZ
Avron The 2016 Nobel prize in Physics: January 2017 17 / 25
The Quantum Hall effect
Chern numbers and Berry’s phase
Stokes∫BZ∇k × A dk1 dk2 =
∮∂BZ
dk · ABZ
Avron The 2016 Nobel prize in Physics: January 2017 17 / 25
The Quantum Hall effect
Chern numbers and Berry’s phase
Stokes∫BZ∇k × A dk1 dk2 =
∮∂BZ
dk · ABZ
Avron The 2016 Nobel prize in Physics: January 2017 17 / 25
The Quantum Hall effect
Chern numbers and Berry’s phase
Stokes∫BZ∇k × A dk1 dk2 =
∮∂BZ
dk · ABZ
Avron The 2016 Nobel prize in Physics: January 2017 17 / 25
The Quantum Hall effect
TKNN aka Chern numbersThe plane is simply connected
T 2 T 2 = R2/Z2∮∂BZ
dk · A = β1 + β2 + β3 + β4 = 2πnj
BZ
β1
β2
β3
β4
Avron The 2016 Nobel prize in Physics: January 2017 18 / 25
The Quantum Hall effect
Aharonov-Bohm ToriDrives and controls
φ2φ1
Driving:emf = φ̇2
Response
Virtual work = − ∂H∂φ1
Avron The 2016 Nobel prize in Physics: January 2017 19 / 25
The Quantum Hall effect
Aharonov-Bohm ToriDrives and controls
φ2φ1
Driving:emf = φ̇2
Response
Virtual work = − ∂H∂φ1
Avron The 2016 Nobel prize in Physics: January 2017 19 / 25
The Quantum Hall effect
Time dependent Feynman-HelmanExpectations related to rates of Berry’s phase
Time and parameter dependent interacting Hamiltonians
i∂t |ψ〉 = H(φ, t) |ψ〉
Quantum observable for virtual work
−∂H∂φ
Expectations related to rate of Berry’s phase〈ψ
∣∣∣∣∂H∂φ∣∣∣∣ψ〉 = i∂t (〈ψ|∂φψ〉)︸ ︷︷ ︸
rate of Berry’s phase
Avron The 2016 Nobel prize in Physics: January 2017 20 / 25
The Quantum Hall effect
Time dependent Feynman-HelmanExpectations related to rates of Berry’s phase
Time and parameter dependent interacting Hamiltonians
i∂t |ψ〉 = H(φ, t) |ψ〉
Quantum observable for virtual work
−∂H∂φ
Expectations related to rate of Berry’s phase〈ψ
∣∣∣∣∂H∂φ∣∣∣∣ψ〉 = i∂t (〈ψ|∂φψ〉)︸ ︷︷ ︸
rate of Berry’s phase
Avron The 2016 Nobel prize in Physics: January 2017 20 / 25
The Quantum Hall effect
Time dependent Feynman-HelmanExpectations related to rates of Berry’s phase
Time and parameter dependent interacting Hamiltonians
i∂t |ψ〉 = H(φ, t) |ψ〉
Quantum observable for virtual work
−∂H∂φ
Expectations related to rate of Berry’s phase〈ψ
∣∣∣∣∂H∂φ∣∣∣∣ψ〉 = i∂t (〈ψ|∂φψ〉)︸ ︷︷ ︸
rate of Berry’s phase
Avron The 2016 Nobel prize in Physics: January 2017 20 / 25
The Quantum Hall effect
Interacting finite systemsQuantized averaged transport
1φ2
time
emf
φ2φ1
Charge transport around loop 1
Q(φ1) =∫ 〈
ψ
∣∣∣∣ ∂H∂φ1∣∣∣∣ψ〉dt
Average adiabatic transport=Chern number∫ 10
Q(φ1)dφ1 = Chern number
Avron The 2016 Nobel prize in Physics: January 2017 21 / 25
The Quantum Hall effect
Interacting finite systemsQuantized averaged transport
1φ2
time
emf
φ2φ1
Charge transport around loop 1
Q(φ1) =∫ 〈
ψ
∣∣∣∣ ∂H∂φ1∣∣∣∣ψ〉dt
Average adiabatic transport=Chern number∫ 10
Q(φ1)dφ1 = Chern number
Avron The 2016 Nobel prize in Physics: January 2017 21 / 25
The Quantum Hall effect
Interacting finite systemsQuantized averaged transport
1φ2
time
emf
φ2φ1
Charge transport around loop 1
Q(φ1) =∫ 〈
ψ
∣∣∣∣ ∂H∂φ1∣∣∣∣ψ〉dt
Average adiabatic transport=Chern number∫ 10
Q(φ1)dφ1 = Chern number
Avron The 2016 Nobel prize in Physics: January 2017 21 / 25
The Quantum Hall effect
Curvature diverges at gap closuresWigner-von Neuman rule
1/B
e2/h
1
2
3
gapped
EF
gapless
EF
i(〈∂1ψ|∂2ψ〉 − 〈∂2ψ|∂1ψ〉
)︸ ︷︷ ︸ill defined at crossing
Avron The 2016 Nobel prize in Physics: January 2017 22 / 25
The Quantum Hall effect
CaveatThermodynamic limit of interacting particles
FQHE: Degenerate ground state (AS, NTW)Disorder(Bellissard, AG, NTW, ASS)Thermodynamics limit(H, BdRF)Open systems (AFG)
φ
EF
Avron The 2016 Nobel prize in Physics: January 2017 23 / 25
The Quantum Hall effect
CaveatThermodynamic limit of interacting particles
FQHE: Degenerate ground state (AS, NTW)Disorder(Bellissard, AG, NTW, ASS)Thermodynamics limit(H, BdRF)Open systems (AFG)
φ
EF
Avron The 2016 Nobel prize in Physics: January 2017 23 / 25
The Quantum Hall effect
CaveatThermodynamic limit of interacting particles
FQHE: Degenerate ground state (AS, NTW)Disorder(Bellissard, AG, NTW, ASS)Thermodynamics limit(H, BdRF)Open systems (AFG)
φ
EF
Avron The 2016 Nobel prize in Physics: January 2017 23 / 25
The Quantum Hall effect
CaveatThermodynamic limit of interacting particles
FQHE: Degenerate ground state (AS, NTW)Disorder(Bellissard, AG, NTW, ASS)Thermodynamics limit(H, BdRF)Open systems (AFG)
φ
EF
Avron The 2016 Nobel prize in Physics: January 2017 23 / 25
The Quantum Hall effect
CaveatThermodynamic limit of interacting particles
FQHE: Degenerate ground state (AS, NTW)Disorder(Bellissard, AG, NTW, ASS)Thermodynamics limit(H, BdRF)Open systems (AFG)
φ
EF
Avron The 2016 Nobel prize in Physics: January 2017 23 / 25
The Quantum Hall effect
CaveatThermodynamic limit of interacting particles
FQHE: Degenerate ground state (AS, NTW)Disorder(Bellissard, AG, NTW, ASS)Thermodynamics limit(H, BdRF)Open systems (AFG)
φ
EF
Avron The 2016 Nobel prize in Physics: January 2017 23 / 25
The Quantum Hall effect
What have TKNN taught us?And what did math-phys contribute?
Non-dissipative transport is quantizedDiophantine equation for Chern number of HofstadterBeautiful geometric picture of adiabatic quantum transportIntroduced Chern numbers & K-theory to CM
Avron The 2016 Nobel prize in Physics: January 2017 24 / 25
The Quantum Hall effect
What have TKNN taught us?And what did math-phys contribute?
Non-dissipative transport is quantizedDiophantine equation for Chern number of HofstadterBeautiful geometric picture of adiabatic quantum transportIntroduced Chern numbers & K-theory to CM
Avron The 2016 Nobel prize in Physics: January 2017 24 / 25
The Quantum Hall effect
What have TKNN taught us?And what did math-phys contribute?
Non-dissipative transport is quantizedDiophantine equation for Chern number of HofstadterBeautiful geometric picture of adiabatic quantum transportIntroduced Chern numbers & K-theory to CM
Avron The 2016 Nobel prize in Physics: January 2017 24 / 25
The Quantum Hall effect
What have TKNN taught us?And what did math-phys contribute?
Non-dissipative transport is quantizedDiophantine equation for Chern number of HofstadterBeautiful geometric picture of adiabatic quantum transportIntroduced Chern numbers & K-theory to CM
Avron The 2016 Nobel prize in Physics: January 2017 24 / 25
The Quantum Hall effect
Hofstadter butterflyA beautiful picture: Avron-Osadchy
Avron The 2016 Nobel prize in Physics: January 2017 25 / 25
David J. ThoulessTKNNThe Quantum Hall effect