weizmann institute of science | - the 2016 nobel prize in physics · 2017. 2. 8. · tknn, simon,...

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

TKNN

TKNN 1982cited 2874

M.V. Berry: Phase ... in adiabatic changes (1984) (cited 7200)B. Simon: Holonomy ... and Berry’s phase (1983)

TKNN

The Classical Hall effect:Ingenious errors 1879

I

V

B

1/B

I/V

T � B

TKNN

I

V

B

1/B

I/V

T � B

TKNN

I

V

B

1/B

I/V

T � B

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

he2≈ 26 [K Ω]

1/B

I/V [e2 h−1]

1.000000000001

1.999999999999

3.000000000006

T � B

he2≈ 26 [K Ω]

1/B

I/V [e2 h−1]

1.000000000001

1.999999999999

3.000000000006

T � B

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

9,192,631,770 [Hz]

Ohm: QHE

I

V

1/B

e2/h

1

2

3

9,192,631,770 [Hz]

Ohm: QHE

I

V

1/B

e2/h

1

2

3

9,192,631,770 [Hz]

Ohm: QHE

I

V

1/B

e2/h

1

2

3

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?

The Problem

1/B

I/V

∣∣∣V =0︸︷︷︸

The Problem

1/B

I/V

∣∣∣V =0︸︷︷︸

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

〈ψ|Lz |ψ〉 ⊆~2R︸︷︷︸

qeqm ∈~c2

∂V 〈ψ |I|ψ〉∣∣∣V =0∈ e

2

hZ

B

〈ψ|Lz |ψ〉 ⊆~2R︸︷︷︸

qeqm ∈~c2

∂V 〈ψ |I|ψ〉∣∣∣V =0∈ e

2

hZ

B

〈ψ|Lz |ψ〉 ⊆~2R︸︷︷︸

qeqm ∈~c2

∂V 〈ψ |I|ψ〉∣∣∣V =0∈ e

2

hZ

B

〈ψ|Lz |ψ〉 ⊆~2R︸︷︷︸

qeqm ∈~c2

∂V 〈ψ |I|ψ〉∣∣∣V =0∈ e

2

hZ

B

〈ψ|Lz |ψ〉 ⊆~2R︸︷︷︸

qeqm ∈~c2

∂V 〈ψ |I|ψ〉∣∣∣V =0∈ e

2

hZ

B

〈ψ|Lz |ψ〉 ⊆~2R︸︷︷︸

qeqm ∈~c2

∂V 〈ψ |I|ψ〉∣∣∣V =0∈ e

2

hZ

B

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

Φ

E

N

Magnetic flux

H = E + N + h.c.

(Nψ)(n,m) = ψ(n,m − 1)

Φ

E

N

Magnetic flux

H = E + N + h.c.

(Nψ)(n,m) = ψ(n,m − 1)

Φ

E

N

Magnetic flux

H = E + N + h.c.

(Nψ)(n,m) = ψ(n,m − 1)

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

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

Gauss Chern and PretzlesFiber bundles for pedestrians

12π

∫Curvature = 2(1− g) ∈ Z

Chern numbersTKNN, Simon, ASS, Bellissard

Bloch Electrons

H(k) = H(k + 2π)

Controlled Hamiltonian

φ2φ1

H(φ) = H(φ+ 2π)

period

+1

-2

Bloch Electrons

H(k) = H(k + 2π)

φ2φ1

H(φ) = H(φ+ 2π)

period

+1

-2

Bloch Electrons

H(k) = H(k + 2π)

φ2φ1

H(φ) = H(φ+ 2π)

period

+1

-2

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

Brillouin zone

EF

Fille

d+1

-1

∂V 〈ψ |I|ψ〉∣∣∣V =0

=e2

hσ︸︷︷︸

Integer

σp = 1 mod q

Brillouin zone

EF

Fille

d+1

-1

∂V 〈ψ |I|ψ〉∣∣∣V =0

=e2

hσ︸︷︷︸

Integer

σp = 1 mod q

Brillouin zone

EF

Fille

d+1

-1

∂V 〈ψ |I|ψ〉∣∣∣V =0

=e2

hσ︸︷︷︸

Integer

σp = 1 mod q

Brillouin zone

EF

Fille

d+1

-1

∂V 〈ψ |I|ψ〉∣∣∣V =0

=e2

hσ︸︷︷︸

Integer

σp = 1 mod q

Hofstadter butterflyTKNN integers everywhere

EF

Φ

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

Period

|ψ〉 eiβ |ψ〉

−π π

Periodic projection

|ψ〉〈ψ| (−π) = |ψ〉〈ψ| (π)

Berry’s phase

|ψ〉 (π) = eiβ |ψ〉 (−π)

β =

∮A(k) · dk

Period

|ψ〉 eiβ |ψ〉

−π π

Periodic projection

|ψ〉〈ψ| (−π) = |ψ〉〈ψ| (π)

Berry’s phase

|ψ〉 (π) = eiβ |ψ〉 (−π)

β =

∮A(k) · dk

Period

|ψ〉 eiβ |ψ〉

−π π

Periodic projection

|ψ〉〈ψ| (−π) = |ψ〉〈ψ| (π)

Berry’s phase

|ψ〉 (π) = eiβ |ψ〉 (−π)

β =

∮A(k) · dk

Period

|ψ〉 eiβ |ψ〉

−π π

Periodic projection

|ψ〉〈ψ| (−π) = |ψ〉〈ψ| (π)

Berry’s phase

|ψ〉 (π) = eiβ |ψ〉 (−π)

β =

∮A(k) · dk

Period

|ψ〉 eiβ |ψ〉

−π π

Periodic projection

|ψ〉〈ψ| (−π) = |ψ〉〈ψ| (π)

Berry’s phase

|ψ〉 (π) = eiβ |ψ〉 (−π)

β =

∮A(k) · dk

Period

|ψ〉 eiβ |ψ〉

−π π

Periodic projection

|ψ〉〈ψ| (−π) = |ψ〉〈ψ| (π)

Berry’s phase

|ψ〉 (π) = eiβ |ψ〉 (−π)

β =

∮A(k) · dk

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

k1,2

EF

Fille

dTKNN

∂V 〈ψ |I|ψ〉∣∣∣V =0

=e2

h×∑

j∈filled

∫BZ

k1,2

EF

Fille

dTKNN

∂V 〈ψ |I|ψ〉∣∣∣V =0

=e2

h×∑

j∈filled

∫BZ

k1,2

EF

Fille

dTKNN

∂V 〈ψ |I|ψ〉∣∣∣V =0

=e2

h×∑

j∈filled

∫BZ

Chern numbers and Berry’s phase

Stokes∫BZ∇k × A dk1 dk2 =

∮∂BZ

dk · ABZ

∮∂BZ

dk · ABZ

∮∂BZ

dk · ABZ

∮∂BZ

dk · ABZ

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

Aharonov-Bohm ToriDrives and controls

φ2φ1

Driving:emf = φ̇2

Response

Virtual work = − ∂H∂φ1

Aharonov-Bohm ToriDrives and controls

φ2φ1

Driving:emf = φ̇2

Response

Virtual work = − ∂H∂φ1

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

i∂t |ψ〉 = H(φ, t) |ψ〉

−∂H∂φ

∣∣∣∣∂H∂φ∣∣∣∣ψ〉 = i∂t (〈ψ|∂φψ〉)︸︷︷︸

i∂t |ψ〉 = H(φ, t) |ψ〉

−∂H∂φ

∣∣∣∣∂H∂φ∣∣∣∣ψ〉 = i∂t (〈ψ|∂φψ〉)︸︷︷︸

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

1φ2

time

emf

φ2φ1

Q(φ1) =∫ 〈

ψ

∣∣∣∣ ∂H∂φ1∣∣∣∣ψ〉dt

1φ2

time

emf

φ2φ1

Q(φ1) =∫ 〈

ψ

∣∣∣∣ ∂H∂φ1∣∣∣∣ψ〉dt

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

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

φ

EF

φ

EF

φ

EF

φ

EF

φ

EF

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

Hofstadter butterflyA beautiful picture: Avron-Osadchy

David J. ThoulessTKNNThe Quantum Hall effect

weizmann institute of science | - the 2016 nobel prize in physics · 2017. 2. 8. · tknn, simon,...

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