Power-law banded random matrices: a testing ground for the Anderson transition

1. Power-law banded random matrices:a testing ground for the Anderson transition Imre Varga Elméleti Fizika Tanszék Budapesti Műszaki és Gazdaságtudományi Egyetem, Magyarország Imre Varga Department of Theoretical Physics Budapest University of Technology and Economics,Hungary collaborators: Daniel Braun (Toulouse) TsampikosKottos (Middletown, CT) José Antonio MéndezBermúdez (Puebla) Stefan Kettemann (Bremen, Pohang), Eduardo Mucciolo (Orlando, FL) thanks to: V.Kravtsov, B.Shapiro, A.Ossipov, A.Garcia-Garcia, Th.Seligman, A.Mirlin, I.Lerner, Y.Fyodorov, F.Evers, B.Eckhardt, U.Smilansky, etc. also to AvH, OTKA, CiC, Conacyt, DFG, etc.

2. Outline • Introduction • Anderson transition • Intermediate statistics • PBRM and the MIT • Spectral statistics, multifractalstates • New results with PBRM at criticality • Scattering • Wave packet dynamics • Entanglement • Magnetic impurities • Summary

3. Anderson model (1958) • Hamiltonian: • Energies enare uncorrelated, random numbers from uniform (bimodal, Gaussian, Cauchy, etc.) distribution  W • Nearest-neighbor hopping  V (symmetry: , , ) • Bloch states for W V, localized states for W  V  W V??

4. One-parameter scaling (1979) Two energy (time) scales:ETh and D (tD and tH) g = ETh/D = tH/tD Gell-Mann – Low function Metal – insulator transition (MIT) for d>2.

6. Mobility edge (d=3) Density of states Conductivity Localized wave functions

7. A non-interacting electron moving in random potential Quantum interference of scattering waves Anderson localization of electrons extended localized localized localized E Ec extended critical

8. Spectral statistics (d=3) MIT Zharekeshev ‘96

9. Spectral statistics (d=3) • W <Wc • extended states • RMT-like • W > Wc • localized states • Poisson-like • W = Wc • multifractal states • intermediate ‘mermaid’

10. Anderson - MIT • Dependence on symmetry parameter  superscalingrelation thru parameter g with and are the RMT limit IV, Hofstetter, Pipek ’99

11. Eigenstates for weak and strong W extended state weak disorder, band center localized state strong disorder, band edge (L=240) R.A.Römer

12. Multifractality at the MIT (3d) Inverse participation numbers • higher accuracy • scaling with L • Box counting technique • fixed L • state-to-state fluctuations http://en.wikipedia.org/wiki/Metal-insulator_transition (L=240) R.A.Römer

13. Multifraktál állapotok a valóságban LDOS fluktuációk a fém-szigetelő átalakulás közelében Ga1-xMnxAs-ban

15. In a metal : anomalous scaling dimensions Multifractality:scaling behavior of moments of (critical) wave functions Critical wave function at a metal-insulator transition point multifractal exponents fractal dimension Continuous set of independent and universal critical exponents singularity spectrum : measure of r where

17. Unusual features of the MIT (3d) • Interplay of eigenvector and spectral statistics • Chalkeret al. ‘95 • Anomalous diffusion at the MIT • Huckesteinet al. ‘97 • Correlation dimension • strong probability overlap (Chalker ’88) • LDOSvs wave function fluctuations • Huckesteinet al. ‘97

18. Unusual features of the MIT (3d) Detect the MIT using a stopwatch! Kottos and Weiss ‘02; Weiss, et al. ‘06

19. PBRM: Power-law Band Random Matrix • Model: matrix with and • asymptotically • parameters: and

20. PBRM • for  RMT, as if • for 1/2 < a < 1  similar to metal with d=1/(a-1/2) • for  BRM Poisson, as if • for a > 3/2 power law localization with exponent a (cf. Yeung-Oono ‘87) • for  criticality (cf. Levitov ‘90) • no mobility edge! • continuous line of transitions: b

21. PBRM transition Cuevas et al.‘01 • asymmetric transition • Kosterlitz-Thouless Kottos and IV ‘01 (unpub.)

22. PBRM at criticality ( ) • for b  1non-linear s-model RG, SUSY(Mirlin‘00) • large conductance: g*=4bb • for b  1 real-space RG, virial expansion, SUSY(Levitov‘99, Yevtushenko-Kravtsov‘03, Yevtushenko-Ossipov ‘07) Mirlin‘00

23. PBRMatcriticality – DOS ( ) b=10.0 b=0.1 b=1.0 L=1024

24. PBRM at criticality (b=1) semi-Poisson statistics is qualitatively valid only IV and Braun ‘00

25. joint distribution state-to-state fluctuation β = 1 β = 2 IV ‘02

26. How does multifractality show up? • Scattering (1 lead) • LDOSvs wave function fluctuations • Anomalous diffusion at the MIT • Nature of entanglement • Screening of magnetic impurities

27. Open system: PBRM + 1 lead • scattering matrix • Wigner delay time • resonance width, eigenvalues of poles of

28. Perfect coupling • distribution of phasesfor b > 1: with • perfect coupling achieved:

29. Scattering: PBRM + 1 lead • JA Méndez-Bermúdez – Kottos ‘05 Ossipov – Fyodorov ‘05: • JA Méndez-Bermúdez – IV06: Measure multifractality using a stopwatch!

30. Wave function and LDOS Wave functions LDOS J.A. Méndez-Bermúdezand IV ‘08 (in prep.)

31. Wave function and LDOS J.A. Méndez-Bermúdezand IV ‘08 (in prep.)

32. Wave packet dynamics survival probability asymptotic wave packet profile J.A. Méndez-Bermúdezand IV ‘08(in prep.)

33. Wave packet dynamics effective dimensionality changes J. A. Méndez-Bermúdezand IV ‘08 (in prep.)

34. B A Entanglement at criticality • 1 qubit in a tight-binding lattice site iwith or without an electron: A • 2 qubits in a tight-binding lattice site iand j with or without an electron: A i • concurrence [Wootters (1997)] (bipartite systems) • tangle [Meyer and Wallach (2002)] (multipartite) i j B A A IV and JAMéndez-Bermúdez ‘08

35. Entanglement at criticality Average concurrence in an eigenstate b=0.3 Average tangle where M=N(N-1)/2and IPR of state IV and JA Méndez-Bermúdez ‘08

36. Entanglement at criticality IV and JAMéndez-Bermúdez ‘08 (cf. Kopp et al. ’07; Jiaet al. ’08)

37. Kondo effektus fémben (1964) T < TKalatt spin-flip szórás, szinglet alapállapot, Kondo-árnyékolás

38. Kondo effektus rendezetlen fémben TK helyfüggő P(TK) széles, bimodális 1-hurok (Nagaoka – Suhl): Árnyékolatlan (szabad) mágneses momentumok, ha Szigetelő: Kissé rendezetlen vezető:

39. Kondo effektus a kritikus pontban lognormális hullámfüggvény eloszlás hullámfüggvény intenzitások együttes eloszlása hullámfüggvények energiakorrelációja

40. Kondo effektus a kritikus pont körül A mágneses momentumok közül pontosan egy szabad: A kritikus pontban nincsenek szabad momentumok A szigetelő oldalon: A kritikus ponttól távolodva léteznek szabad momentumok

41. A fém-szigetelő átalakulás szimmetriája Kritikus pont szimmetria függő: esetén

42. Magnetic impurity S Kettemann, E Mucciolo, IV ‘09

43.   

44. Summary • PBRM: a good testing ground for the Anderson transition • d=1→ scaling with L • no mobility edge (!) • features similar to Anderson MIT → deviations found • tunable transition → bserve as 1/d or g • multifractal states induce unusual behavior • Scattering • Wave packet dynamics • Entanglement • Interplay with magnetic impurities • Outlook • Effect of interactions on the HF level • Dynamical stability versus chaotic environment Thankyouforyourattention

45. Outlook: Current and future problems • free magnetic moments + e-e interactions • S. Kettemann (Hamburg) • E. Mucciolo(Orlando) • interplay of multifractality and interaction • decoherence of qubits in critical environment • Th. Seligman (Cuernavaca) • J.A. Méndez-Bermúdez (Puebla)