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On the Multifractal Dimensions at the Anderson Transition

On the Multifractal Dimensions at the Anderson Transition. Imre Varga Departament de Física Teòrica Universitat de Budapest de Tecnologia i Economia , Hongria. Imre Varga Elm életi Fizika Tanszék Budapesti Műszaki és Gazdaságtudományi Egyetem, H-1111 Budapest, Magyarország.

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On the Multifractal Dimensions at the Anderson Transition

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  1. On the Multifractal Dimensionsat the AndersonTransition Imre Varga Departament de Física Teòrica Universitat de Budapest de Tecnologiai Economia, Hongria Imre Varga Elméleti Fizika Tanszék Budapesti Műszaki és Gazdaságtudományi Egyetem, H-1111 Budapest, Magyarország Coauthors: José Antonio Méndez-Bermúdez, AmandoAlcázar-López (BUAP, Puebla, México) thanks to : OTKA, AvH

  2. Outline • Introduction • The Anderson transition • Essential features of multifractality • Random matrix model: PBRM • Heuristic relations forgeneralizeddimensions • Spectral compressibility vs. multifractality • Wigner-Smith delay time • Further tests • Conclusions and outlook

  3. Anderson’s model (1958) • Hamiltonian • Energiesenuncorrelated, randomnumbers from uniform(bimodal, Gaussian, Cauchy, etc.) distribution W • Nearest-neighbor„hopping” V (symmetries:R, C, Q) • Bloch states for W V, localized states for W  V W V?

  4. Anderson localization Sridhar 2000 Billyet al. 2008 Jendrzejewskiet al. 2012 Hu et al. 2008

  5. Spectral statistics • W <Wc • extended states • RMT-like: • W > Wc • localizedstates • Poisson-like: • W = Wc • multifractalstates • intermediatestat. ‘mermaid’ semi-Poisson RMT

  6. Eigenstates at small and largeW Extended state Weak disorder, midband Localized state Strong disorder, bandedge (L=240) R.A.Römer

  7. Multifractal eigenstate at the critical point http://en.wikipedia.org/wiki/Metal-insulator_transition (L=240) R.A.Römer

  8. Multifractal eigenstate at the critical point • Inverse participation ratio • higher precision • scaling withL • Box-counting technique • fixedL • „state-to-state”fluctuations • PDF analyzis

  9. Multifractal eigenstate at the critical point Do these states exist at all? Yes

  10. Multifractal states in reality LDOS változása a QH átmeneten keresztül n-InAs(100) felületre elhelyezett Cs réteggel

  11. Multifractal states in reality LDOS fluctuations in the vicinity of the metal-insulator transition Ga1-xMnxAs

  12. Multifractality in general • Turbulence (Mandelbrot) • Time series – signal analysis • Earthquakes • ECG, EEG • Internet data traffic modelling • Share, asset dynamics • Music sequences • etc. • Complexity • Human genome • Strange attractors • etc. Common features • self-similarity across many scales, • broad PDF • muliplicative processes • rare events

  13. Multifractality in general • Very few analytically known • binary branching process • 1d off-diagonal Fibonacci sequence • Baker’s map • etc • Numerical simulations • Perturbation series (Giraud 2013) • Renormalization group - NLM – SUSY (Mirlin) • Heuristic arguments

  14. Numerical multifractal analysis Parametrization of wave function intensities Theset of pointswherescales with Scaling: • Box size: • System size: Averaging: • Typical: • Ensemble:

  15. Numerical multifractal analysis Parametrization of wave function intensities Theset of pointswherescales with • convex • Symmetry (Mirlin, et al. 06)

  16. Numerical multifractal analysis Generalized inverse participation number, Rényi-entropies Mass exponent, generalized dimensions Wave function statistics parabolic og-normal

  17. Numerical multifractal analysis applicationtoquantumpercolation, seeposterby L. Ujfalusi Rodriguezet al. 2010

  18. Correlations at the transition Interplay of eigenvector and spectralcorrelations • q=2, Chalker et al. 1995 • q=1, Bogomolny 2011 Cross-correlation of multifractaleigenstates Auto-correlation of multifractaleigenstates Enhanced SC Feigel’man2007 Burmistrov 2011 New Kondophase Kettemann 2007

  19. Effect of multifractality(PBRM) Generalize! Take the model of the model! PBRM (a random matrix model)

  20. b PBRM: Power-law Band Random Matrix • model: matrix, • asymptotically: • free parameters and Mirlin, et al. ‘96, Mirlin ‘00

  21. PBRM Mirlin, et al. ‘96, Mirlin ‘00

  22. weakmultifractality • strong multifractality Mirlin, et al. ‘96, Mirlin ‘00

  23. Generalizeddimensions JAMB és IV (2012)

  24. PBRM atcriticality General relations Spectralstatistics and e.g.: JAMB és IV (2012)

  25. Higher dimensions, 2dQHT Replace For using 3dAMIT JAMB és IV (2012)

  26. Surpisingly robust and general Different problems • Random matrix ensembles • Ruijsenaars-Schneider ensemble • Critical ultrametric ensemble • Intermediate quantum maps • Calogero-Moser ensemble • Chaos • baker’s map • Exact, deterministic problems • Binary branching sequence • Off-diagonal Fibonacci sequence JAMB és IV (2013)

  27. Scattering: system + lead • Scatteringmatrix • Wigner-Smith delay time • Resonance widths: eigenvalues of polesof

  28. Scattering: PBRM + 1 lead • JA Méndez-Bermúdez – Kottos ‘05 Ossipov – Fyodorov ‘05: • JA Méndez-Bermúdez – IV06:

  29. Scattering exponents Wigner-Smith delay time JAMB és IV (2013)

  30. Summary and outlook • Multifractal states in general • Randommatrixmodel (PBRM) • heuristic relations tested for many models, quantities • New physicsinvolved • Kondo, SC, graphene, etc. • Outlook • Interactingparticles (cf. Mirlinet al. 2013) • Decoherence • Proximityeffect (SC) • Topologicalinsulators Thanks for your attention

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