The 2016 Nobel prize in Physics D. Thouless and Topological Invariants J. Avron May 2017 Avron The 2016 Nobel prize in Physics: May 2017 1 / 24
There is geometry in the humming of the strings, there is music in the spacing of the spheres. (Pythagoras) Happy birthday, Petr Avron The 2016 Nobel prize in Physics: May 2017 2 / 24
D. Thouless, D. Haldane, M. Kosteritz Kosterlitz-Thouless transition; TKNN aka Chern numbers Avron The 2016 Nobel prize in Physics: May 2017 3 / 24
Mathematical physics in 2 D Kosterlitz-Thouless transition; TKNN aka Chern numbers Quantum transistors Marginal phase transition Avron The 2016 Nobel prize in Physics: May 2017 4 / 24
TKNN 1982 cited 2874 TKNN: Topological quantum numbers (1982) B. Simon: Chern classes in QM (1983) M.V. Berry: Adiabatic curvature, Berry’s phase (1984) Avron The 2016 Nobel prize in Physics: May 2017 5 / 24
Maxwell’s Ingenious blunders Avron The 2016 Nobel prize in Physics: May 2017 6 / 24
The Classical Hall effect 1879 V I / V T ≫ B B I 1 / B Avron The 2016 Nobel prize in Physics: May 2017 7 / 24
The Quantum Hall effect The Quantum Hall effect von Klitzing (Nobel 1985) Quantum unit of resistance h e 2 ≈ 26 [ K Ω] I / V [ e 2 h − 1 ] 3 . 000000000006 1 . 999999999999 T ≪ B 1 . 000000000001 1 / B Avron The 2016 Nobel prize in Physics: May 2017 8 / 24
The Quantum Hall effect Fundamental vs natural standards Time: Natural but not fundamental Second: Hyperfine transition of Cs 133 9 , 192 , 631 , 770 [ Hz ] All Cs 133 atoms are equal Natural, NOT fundamental 9 , 192 , 631 , 770 [ Hz ] is precisely measurable, but Not related to a fundamental time scale in a known way Avron The 2016 Nobel prize in Physics: May 2017 9 / 24
The Quantum Hall effect Fundamental vs natural standards Ohm: Artificial but fundamental Ohm: QHE e 2 / h V 3 I 2 1 1 / B Every transistor is different Artificial but fundamental Resistance: The quantum unit of resistance is precisely measurable Avron The 2016 Nobel prize in Physics: May 2017 10 / 24
The Quantum Hall effect Heisenberg quantization Quantization and spectral theory Observables=Linear operators Measurements yield eigenvalues Spect ( L z ) ⊆ � 2 Z Not the mechanism in the QHE Avron The 2016 Nobel prize in Physics: May 2017 11 / 24
The Quantum Hall effect Dirac quantization Electric charges are an integer Electric charges ∈ q e Z q electron = − q proton Magnetic monopole q mag B q electric q mag ∈ � c 2 Z Not the mechanism in the QHE Avron The 2016 Nobel prize in Physics: May 2017 12 / 24
The Quantum Hall effect TKNN quantization Quantization of transport coefficients Topological invariants & Transport Hall Conductance= Chern number Gauss-Bonnet-Chern � 1 Curvature ∈ Z 2 π Avron The 2016 Nobel prize in Physics: May 2017 13 / 24
The Quantum Hall effect Real scientists solve models. Wimps generalize M. Berry The Hofstadter model ψ ( n , m ) ∈ ℓ 2 ( Z 2 ) B North translation � � N N ψ ( n , m ) = ψ ( n , m − 1 ) E East translations: � � ( n , m ) = e − 2 π iB m ψ ( n − 1 , m ) E ψ Hamiltonian H = E + N + h . c . Avron The 2016 Nobel prize in Physics: May 2017 14 / 24
The Quantum Hall effect Periodic matrices The importance of families When B = p q : H is periodic and reduces to = e ik 1 + e ik 2 ˆ H ( k 1 , k 2 ) T T + h . c . ���� ���� � �� � cyclic shift FT of cyclic shift q × q periodic matrix Example B = 1 3 : 0 1 0 1 0 0 ˆ , ω = e 2 π iB T = , T = 0 ω 0 0 0 1 � �� � ω 2 1 0 0 0 0 root of unity � �� � 3 × 3 Avron The 2016 Nobel prize in Physics: May 2017 15 / 24
The Quantum Hall effect Quantum states as bundles of projections Full bands of free Fermions Spectrum H ( k 1 , k 2 ) : periodic q × q hermitian matrix Spectrum ( H ) : q-bands. P j ( k 1 , k 2 ) : bundles of projections P 2 Quantum states at T = 0 Finite system: Rank 1 projection Full bands of free Fermions: Bundle P 1 of projections k Avron The 2016 Nobel prize in Physics: May 2017 16 / 24
The Quantum Hall effect Families of spectral projections H ( k 1 , k 2 ) P ( k 1 , k 2 ) Smooth, periodic spectral projection P ( k 1 , k 2 ) = P ( k 1 + 2 π, k 2 ) = P ( k 1 , k 2 + 2 π ) Rank one projection: P = | ψ � � ψ | � �� � family Avron The 2016 Nobel prize in Physics: May 2017 17 / 24
The Quantum Hall effect Curvature in differential geometry Curvature: Failure of parallel transport Z Curvature Failure of parallel transport area 4 π R 2 / 8 = 1 π/ 2 X Y R 2 Avron The 2016 Nobel prize in Physics: May 2017 18 / 24
The Quantum Hall effect Curvature of bundles of projections in Hilbert space Berry’s phase: Failure of parallel transport | ψ � Berry’s (gauge) 1-form A = i � ψ | d k ψ � Berry’s phase: Failure of parallel transport | ψ � � � A = dA P � � Curvature: Local failure of parallel transport d A ⇐ ⇒ ( dA ) jk ( φ ) = − 2 Im � ∂ j ψ | ∂ k ψ � ���� 2 − form Avron The 2016 Nobel prize in Physics: May 2017 19 / 24
The Quantum Hall effect Expectation of currents=Rates of Berry’s gauge Evolution equation i d | ψ t � = H ( k , t ) | ψ t � dt Define current: ∂ H ∂ k Expectations related to rate of Berry’s phase � � � � ∂ H i d � � ψ t � ψ t = dt � ψ | ∂ k ψ � � � ∂ k � � �� � � �� � rate of Berry’s phase expectation of current Avron The 2016 Nobel prize in Physics: May 2017 20 / 24
The Quantum Hall effect Hall conductance=Chern number TKNN � Hall conductance = 1 T 2 dA 2 π Gauss-Bonnet-Chern rediscovered T 2 = R 2 / Z 2 � 1 T 2 dA ∈ Z 2 π β 3 � � β 4 T 2 β 2 T 2 dA = ∂ T 2 A = β 1 + β 2 + β 3 + β 4 ∈ 2 π Z β 1 Avron The 2016 Nobel prize in Physics: May 2017 21 / 24
The Quantum Hall effect Hofstadter butterfly Fractal diagram of Chern numbers B density Avron The 2016 Nobel prize in Physics: May 2017 22 / 24
The Quantum Hall effect What have we learned? And what did I not cover Transport coefficients have geometric significance Macroscopic systems of Fermions are bundles of projections Bundles of projections are related to Chern classes K-theory, Entanglement, Topological states of matter,.... Avron The 2016 Nobel prize in Physics: May 2017 23 / 24
The Quantum Hall effect 2016 Nobel prize food for thought Fiber bundles with cream cheese Avron The 2016 Nobel prize in Physics: May 2017 24 / 24
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