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1d Anderson localization: devil’s staircase of statistical anomalies

1d Anderson localization: devil’s staircase of statistical anomalies. V.E.Kravtsov, Abdus Salam ICTP, Trieste & Landau Institute. Collaboration: V.I.Yudson. Discussion: A.Ossipov. arXiv:0806.2118. H. E F , f. F. Hofstadter butterfly: hierarchy of spectral gaps.

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1d Anderson localization: devil’s staircase of statistical anomalies

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  1. 1d Anderson localization: devil’s staircase of statistical anomalies V.E.Kravtsov, Abdus Salam ICTP, Trieste & Landau Institute Collaboration: V.I.Yudson Discussion: A.Ossipov arXiv:0806.2118

  2. H EF, f F Hofstadter butterfly: hierarchy of spectral gaps Harper, Thouless… Wigmann &Zabrodin Magnetic field opens up spectral gaps in a 2D lattice tight-binding model: effect of commensurabilityof magnetic flux through a unit cell and the magnetic flux quantum

  3. l E The width of anomalous region At E=0 (f=1/2): At E=1 (f=1/3 or 2/3): Anderson localization: hierarchy of statistical anomalies a F.Wegner, 1980; Derrida & Gardner, 1986 Titov, 2002 Deytch et al 2003 Effect of commensurability of the lattice constant a and the Fermi-wavelength l. This effect is not present in the continuous model. Localization length sharply increases in the vicinity of E=0 and sharply decreases in the vicinity of E=1

  4. ? What about the entire wavefunction statistics?

  5. Generalized Fokker-Planck equation Result of the super-symmetric quasi-sigma model (Ossipov, Kravtsov 2006) The function F is found from the generalized Fokker-Planck equation: where Is the localization length without anomalous contributions

  6. Where are the anomalous terms? The term has a part that emerges only at f=m/(p+2)

  7. Derivation…

  8. …derivation

  9. This term survives in the continuous limit This term is present only for f=1/2 Center of the band anomaly f=1/2 Dependence of the Fokker-Plank equation on the phase at f=1/2 Is this a mess?

  10. Exact integrability of the Fokker-Plank equation N-> infinity: obeys zero-mode Fokker-Planck Equation This equation is exactly integrable

  11. Separation of variables and the inverse-square Hamiltonian Celebrated inverse-square Hamiltonian

  12. Problem of continuous degeneracy W is the Whittaker function Q: How to fix the function C(l)? A: Smoothness of F(u,f) with all the derivatives + some miraculous properties of the Whittaker function

  13. Final result for the generating function WHY SO MUCH OF A MIRACLE? WHAT IS THE HIDDEN SYMMETRY? What is the new properties of the anomalous statistics?

  14. Conclusion • Statistical anomalies in 1d Anderson model at any rational filling factor f. • Integrability of the TM equation for GF determining all local statistics at a principal anomaly at f=1/2. • Unique solution for the GF in the infinite 1d Anderson model. • Hidden symmetry that makes TM equation integrable?

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