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Imre Varga Elméleti Fizika Tanszék Budapesti Műszaki és Gazdaságtudományi Egyetem, H-1111 Budapest, Magyarország Imre Varga Departament de Física Teòrica Universitat de Budapest de Tecnologia i Economia, Hongria On the Multifractal Dimensions at the Anderson Transition Coauthors: José Antonio Méndez-Bermúdez, Amando Alcázar-López (BUAP, Puebla, México) thanks to : OTKA, AvH
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Outline Introduction The Anderson transition Essential features of multifractality Random matrix model: PBRM Heuristic relations for generalized dimensions Spectral compressibility vs. multifractality Wigner-Smith delay time Further tests Conclusions and outlook
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Anderson’s model (1958) Hamiltonian Energies e n uncorrelated, random numbers 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 ?
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Anderson localization Billy et al. 2008 Hu et al. 2008 Sridhar 2000 Jendrzejewski et al. 2012
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Spectral statistics
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Eigenstates at small and large W Extended state Weak disorder, midband Localized state Strong disorder, bandedge (L=240) R.A.Römer
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Multifractal eigenstate at the critical point http://en.wikipedia.org/wiki/Metal-insulator_transitionhttp://en.wikipedia.org/wiki/Metal-insulator_transition (L=240) R.A.Römer http://en.wikipedia.org/wiki/Metal-insulator_transitionhttp://en.wikipedia.org/wiki/Metal-insulator_transition (L=240) R.A.Römer
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Multifractal eigenstate at the critical point Inverse participation ratio Box-counting technique fixed L „state-to-state” fluctuations PDF analyzis higher precision scaling with L
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Multifractal eigenstate at the critical point
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Do these states exist at all?
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Multifractal states in reality LDOS változása a QH átmeneten keresztül n-InAs(100) felületre elhelyezett Cs réteggel
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Multifractal states in reality LDOS fluctuations in the vicinity of the metal-insulator transition Ga 1-x Mn x As
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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 Common features self-similarity across many scales, broad PDF muliplicative processes rare events
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Multifractality in general
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Numerical multifractal analysis Parametrization of wave function intensities
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Numerical multifractal analysis Parametrization of wave function intensities
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Numerical multifractal analysis Generalized inverse participation number, Rényi-entropies Mass exponent, generalized dimensions Wave function statistics
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Numerical multifractal analysis Rodriguez et al. 2010application to quantum percolation, see poster by L. Ujfalusi
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Correlations at the transition Interplay of eigenvector and spectral correlations q=2, Chalker et al. 1995 q=1, Bogomolny 2011 Cross-correlation of multifractal eigenstates Auto-correlation of multifractal eigenstates
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Effect of multifractality (PBRM) Generalize! Take the model of the model! PBRM (a random matrix model)
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b PBRM: Power-law Band Random Matrix model:matrix, asymptotically: free parameters and Mirlin, et al. ‘96, Mirlin ‘00
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PBRM Mirlin, et al. ‘96, Mirlin ‘00
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JAMB és IV (2012) Generalized dimensions
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JAMB és IV (2012) General relations e.g.:
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JAMB és IV (2012) Replace 2dQHT 3dAMIT
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JAMB és IV (2013) 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 Surpisingly robust and general
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Scattering: system + lead Scattering matrix Wigner-Smith delay time Resonance widths: eigenvalues of poles of
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JA Méndez-Bermúdez – Kottos ‘05 Ossipov – Fyodorov ‘05: JA Méndez-Bermúdez – IV 06: Scattering: PBRM + 1 lead
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JAMB és IV (2013) Wigner-Smith delay time Scattering exponents
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Summary and outlook Multifractal states in general Random matrix model (PBRM) heuristic relations tested for many models, quantities New physics involved Kondo, SC, graphene, etc. Outlook Interacting particles ( cf. Mirlin et al. 2013 ) Decoherence Proximity effect (SC) Topological insulators Thanks for your attention
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Energy (time) scales: E Th and D ( t D and t H ) g = E th / D = t H / t D One-parameter scaling (1979) MIT: d>2 (O), d>1 (Sp) Gell-Mann – Low function n O16.451.57 U18.321.43 Sp18.931.375
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