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Quantum Hall effect in the presence of magnetic impurities

Quantum Hall effect in the presence of magnetic impurities. V. Kagalovsky Shamoon College of Engineering Beer- Sheva , Israel. TIDS 15. Context. Integer quantum Hall effect, semiclassical picture & Chalker-Coddington network model

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Quantum Hall effect in the presence of magnetic impurities

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  1. Quantum Hall effect in the presence of magnetic impurities V. Kagalovsky Shamoon College of Engineering Beer-Sheva, Israel TIDS 15

  2. Context Integer quantum Hall effect, semiclassical picture & Chalker-Coddington network model Spin-flip tunneling – exact solution Averaging over magnetic impurities’ spins – effective scattering matrix for an electron only Nonunitarity as a measure of decoherence

  3. Semiclassical picture: strong magnetic field B and slowly varying random potential V(r) Electron moves alonglines of constant potential Transmission probability Scattering in the vicinity of the saddle point potential Network model – numerical calculations – localization

  4. The network model of Chalker and Coddington. Each node represents a saddle point and each link an equipotential line of the random potential (Chalker and Coddington; 1988) Isotropy The most “quantum” case Crit. value argument

  5. Fertig and Halperin, PRB 36, 7969 (1987) Exact transmission probability through the saddle-point potential in the presence of perpendicular magnetic field for strong magnetic fields For the network model

  6. Total transfer matrix T of the system is a result of N iterations. Real parts of the eigenvalues are produced by diagonalization of the product M – system widthLyapunov exponents1>2>…>M/2>0 Localization length for the system of width M M is related to the smallest positive Lyapunov exponent: M ~ 1/M/2 Loc. Length explanation

  7. The effect of nuclear spin on the tunneling at the node 3 1 2 4

  8. dimensionless strength of the Zeeman coupling relative exchange strength

  9. 3 1 4 2

  10. 3 1 2 4 V. K. and A. L. Chudnovskiy, J. Mod. Phys. (2011). Vol. 2 pp. 970-976

  11. “Spin-flip” tunneling “No spin-flip” tunneling

  12. Can we average over nuclear spin to get an effective scattering matrix for an electron only? 3 1 4 2

  13. Density matrix of a nuclear spin before electron arrives Quantum statistical averaging of an observable A

  14. Quantum probability to for an electron to be reflected Probability to for an electron to be reflected averaged over nuclear spin

  15. The probability matrix Zeeman splitting

  16. 4x4 matrix of statistically averaged probabilities to tunnel and to be reflected SPsatisfies current conservation. e.g. for an incoming electron the total probability is

  17. Effective scattering matrix for an electron has to satisfy Unitarity of the scattering matrix implies But the probability matrix satisfies only because any spin which enters must either tunnel or be reflected with or without spin-flip

  18. The second condition completely unpolarized magnetic impurity is satisfied only for Averaging over nuclear spins effectively breaks TR. TR of quantum mechanics is restored because statistical process is reversible (entropy is maximal) For

  19. The probability matrix Zeeman splitting

  20. But this not enough!!! Decoherence is introduced by averaging over magnetic impurity.In fictitious scattering matrix rows and columns are not orthogonal even for the completely unpolarized magnetic impurities What can be done?

  21. Toy model Two incoming states Two outgoing states The degree of decoherence grows with The eigenvalues of both

  22. Back to the fictitious scattering matrix

  23. Nonunitarity but their eigenvalues coincide, are degenerate and equal to 1+c or 1-c. So, one parameter cdescribes the deviation from 1 (UNITARITY) – Toy model it can be considered as a measure of decoherence!!! Exact solution

  24. For weak spin-spin exchange strength d<<1 For finite polarization For completely unpolarized impurity

  25. Phase uncertainty after a single scattering event The total phase uncertainty after N scattering events is evaluated as a sum of random phases Electron is completely incoherent then after scattering events Elliot – Yafet mechanism

  26. Phase coherence time Impurity concentration for completely unpolarized impurity for finite polarization

  27. Finite width of the inter-plateaux transition in the quantum Hall effect due to magnetic impurities Phase coherence length Localization length The condition defines the width of the transition V. K. and A. L. Chudnovskiy, JETP (2013). Vol. 116 pp. 657- 662

  28. Summary Importance of the spin-spin interactions in the quantum Hall effect! Nonunitarity of Scattering Matrix as Signature of Decoherence

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