1. 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

2. 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

3. 

4. 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 ?

5. Anderson localization Billy et al. 2008 Hu et al. 2008 Sridhar 2000 Jendrzejewski et al. 2012

6. Spectral statistics

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

8. 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

9. 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

10. Multifractal eigenstate at the critical point

12. Do these states exist at all?

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

14. Multifractal states in reality LDOS fluctuations in the vicinity of the metal-insulator transition Ga 1-x Mn x As

15. 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

16. Multifractality in general

17. Numerical multifractal analysis Parametrization of wave function intensities

18. Numerical multifractal analysis Parametrization of wave function intensities

19. Numerical multifractal analysis Generalized inverse participation number, Rényi-entropies Mass exponent, generalized dimensions Wave function statistics

20. Numerical multifractal analysis Rodriguez et al. 2010application to quantum percolation, see poster by L. Ujfalusi

21. 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

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

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

24. PBRM  Mirlin, et al. ‘96, Mirlin ‘00

26. JAMB és IV (2012) Generalized dimensions

27. JAMB és IV (2012) General relations e.g.:

28. JAMB és IV (2012) Replace 2dQHT 3dAMIT

29. 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

30. Scattering: system + lead  Scattering matrix  Wigner-Smith delay time  Resonance widths: eigenvalues of poles of

31.  JA Méndez-Bermúdez – Kottos ‘05 Ossipov – Fyodorov ‘05:  JA Méndez-Bermúdez – IV 06: Scattering: PBRM + 1 lead

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

33. 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

34. 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