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Quantum Hall Effect Josephson Effect Radiative Corrections to Quantum Hall and Josephson Effects Alexander Penin University of Alberta & INR Moscow RADCOR 2009 25 - 30 October 2009, Ascona, Switzerland A. Penin, U of A & INR RADCOR 2009 – p. 1/16 Topics Discussed Exact results in quantum mechanics Flux quantization Quantum Hall effect Josephson effect Vacuum polarization in strong magnetic field QED corrections to quantum Hall and Josephson effect Theory Experiment A. Penin, U of A & INR RADCOR 2009 – p. 2/16 Exact result: an example Determination of the fine structure constant α electron g − 2 10 experimental error: 10− 10 theoretical error: 10− - requires 4-loop calculation in QED quantum Hall effect 8 12 experimental error: 10− (could be 10− ) theoretical error: 0 - for free! A. Penin, U of A & INR RADCOR 2009 – p. 3/16 Gauge invariance and quantum phase 2 ~ e = c = 1, α = 4π ✔ Schrödinger equation (i∂t − H)Ψ=0 ✔ Wave function Ψ = |Ψ|eiθ ✔ Hamiltonian H = qA0 + F (B, E, D) D = ∂ − iqA r1 ➥ If ∇ × A = 0 then θ(r1) − θ(r2) = q A(r′) · dr′ Zr2 A. Penin, U of A & INR RADCOR 2009 – p. 4/16 Flux quantization F. London (1948) Superconductivity ➪ coherent state of Cooper pairs ➪ q = 2e Meissner effect ➪ B = 0 inside superconductor Superconducting ring: × S B C ➪ Single-valued wave function ∆θ = 2e C A · dx = 2πn H A. Penin, U of A & INR RADCOR 2009 – p. 5/16 Flux quantization F. London (1948) Superconductivity ➪ coherent state of Cooper pairs ➪ q = 2e Meissner effect ➪ B = 0 inside superconductor Superconducting ring: × S B C ➪ Single-valued wave function ∆θ = 2e C A · dx = 2πn 2 ➪ πn C A(r) · r = S B(r) · d s ≡ Φ Φ = e ≡H nΦ0 H R h Flux quantum: Φ0 = 2e A. Penin, U of A & INR RADCOR 2009 – p. 6/16 Hall effect E. Hall (1879) B I Lorentz force vs electrostatic force z x eE = −ev × B y E ρe current density j = ρev = E B ρe total current per length I = V B 1 ρe Hall conductivity R− = B A. Penin, U of A & INR RADCOR 2009 – p. 7/16 Quantum Hall effect K. von Klitzing, G. Dorda, M. Pepper (1980) Wave function: 2 x i πm L L Ψ(x,y) = e ψ(y − ym) ψ(y − ym) ➪ harmonic oscillator 2πm centered at ym = eBL 2π 1 eBL √eB Density of quantum states with eB n Landau levels filled: ρ = n 2π 1 Quantum Hall conductivity: R− = 2nα = n/RK h von Klitzing constant: RK = e2 A. Penin, U of A & INR RADCOR 2009 – p. 8/16 Gauge invariance argument R.B. Laughlin (1981) δH B current density j ∝ E δA dE I total current I = dΦ Φ Flux quantization Φ = 4π|A|/L = n2Φ0 Flux dependence ym(Φ + 2Φ0) = ym+1(Φ) for dΦ = 2Φ0 ➪ dE = neV neV nV ➥ I = = 2Φ0 RK A. Penin, U of A & INR RADCOR 2009 – p. 9/16 Vacuum polarization × J. Schwinger (1951) = + +... q → × 2 Charge renormalization δe ∝ Πµν(q)/q |q2 0 → Vacuum polarization in magnetic field 2 α eB 1 2 2 2 δΠµν (q)= − 2 gµν q − qµqν − 7 gµν q − qµqν + 4 gµν q − qµqν π m2 45 k ⊥ Lorentz invariance broken ➪ different couplings A. Penin, U of A & INR RADCOR 2009 – p. 10/16 QED corrections to electromagnetic couplings Coulomb potential 3 2 2 δΠ00(q) −iq·r d q α α eB 2 7 2 VC (r)= e 1 − e = 1+ − sin θ q2 (2π)3 |r| π m2 45 90 Z " # Interaction with the homogeneous electric field 2 α eB 11 1+ e r·E ≡ e∗r·E π m2 180 ! Hall current 2 α eB 1 e′′ 1+ I ≡ I π m2 15 e ! A. Penin, U of A & INR RADCOR 2009 – p. 11/16 QED corrections to RK A.P., Phys. Rev. B 79, 113303 (2009) 1 2 Origin of the corrections RK− ∝ e → e∗e′′ Result: 2 2 2 2 1 e 23 α ~eB e 20 B R− = 1 + ≈ 1 + 10− × K h 180 π c2m2 h 10 T " # " # A. Penin, U of A & INR RADCOR 2009 – p. 12/16 Josephson effect B.D. Josephson (1962); S. Shapiro (1963) Tunneling Josephson current |Ψ|eiθ1 I ≈ Ic sin(θ1 − θ2) E B Constant voltage V across the junction |Ψ|eiθ2 ➥ the current oscillates at frequency ν Josephson frequency-voltage relation ν = KJ V 2e Josephson constant KJ = h A. Penin, U of A & INR RADCOR 2009 – p. 13/16 QED corrections to KJ A.P. (preliminary result) Origin of the corrections KJ ∝ e → e∗ Result: 2 2 2e 11 α ~eB 2e 20 B K = 1 + ≈ 1 + 10− × J h 180 π c2m2 h 10 T " # " # A. Penin, U of A & INR RADCOR 2009 – p. 14/16 Experiment Quantum Hall effect (F. Schopfer, W. Poirier (2007)) thermal Johnson-Nyquist noise ➪ 12 ✘ accuracuy limit 10− Josephson effect (A.K. Jain, J.E. Lukens, J.-S. Tsai (1987)) 19 ✔ accuracy 10− A. Penin, U of A & INR RADCOR 2009 – p. 15/16 Summary Vacuum polarization results in a weak dependence of the Josephson and von Klitzing constant on the magnetic field strength This remarkable manifestation of a fine nonlinear quantum field effect in collective phenomena in condensed matter can be observed in a dedicated experiment One literally can measure the vacuum polarization with a voltmeter! A. Penin, U of A & INR RADCOR 2009 – p. 16/16.
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