Oleg Yevtushenko

1. LMU Critical Propagators in Power Law Banded RMT:from multifractality to Lévy flights Oleg Yevtushenko In collaboration with: Philipp Snajberk (LMU & ASC, Munich) Vladimir Kravtsov (ICTP, Trieste)

2. Outline of the talk • Introduction: Unconventional RMT with fractal eigenstates  Multifractality and Scaling properties of correlation functions • Strong multifractality regime: •  Basic ideas of SuSy Virial Expansion •  Application of SuSyVE for generalized diffusion propagator • Scaling of retarded propagator: Results and Discussion: multifractality, Lévy flights, phase correlations • Conclusions and open questions Bielefeld, 16 December 2011

3. WD and Unconventional Gaussian RMT The Schrödinger equation for a 1d chain:(eigenvalue/eigenvector problem) Statistics of RM entries: If F(i-j)=1/2→ the Wigner–Dyson (conventional) RMT (1958,1962): Parameter βreflects symmetry classes(β=1: GOE, β=2: GUE) Generic unconventional RMT: F(x) A x 1 FunctionF(i-j) can yield universality classes of the eigenstates, different from WD RMT Factor A parameterizes spectral statistics and statistics of eigenstates H-Hermithian matrix with random (independent, Gaussian-distributed) entries

4. 3 cases which are important for physical applications localized extended (WD) k k Fractal eigenstates Model for systems at the critical point Model for insulators Model for metals Inverse Participation Ratio (fractal) dimension of a support: the space dimension d=1 for RMT

5. MF RMT: Power-Law-Banded Random Matrices 2 π b>>1 b<<1 bis the bandwidth b 1-d2<<1 – regime of weak multifractality d2<<1 – regime of strong multifractality RMT with multifractal eignestates at any band-width (Mirlin, Fyodorov et.al., 1996, Mirlin, Evers, 2000 - GOE and GUE symmetry classes)

6. Correlations of the fractal eigenstates Two point correlation function: Critical correlations(Wegner, 1985; Chalker, Daniel, 1988; Chalker 1980) ω  Lω L If ω> then must play a role of L: For a disordered system at the critical point (fractal eigenstates)

7. Correlations of the fractal eigenstates Two point correlation function: Critical correlations(Wegner, 1985; Chalker, Daniel, 1988; Chalker 1980) For a disordered system at the critical point (fractal eigenstates) Dynamical scaling hypothesis: d –space dimension,  - mean level spacing, l – mean free path, <…> - disorder averaging

8. Fractal enhancement of correlations Extended:small amplitude substantial overlap in space Localized:high amplitude small overlap in space localized critical extended Fractal: relatively high amplitude and the fractal wavefunctions strongly overlap in space (Cuevas , Kravtsov, 2007) Dynamical scaling: - Enhancement of correlations (The Anderson model: tight binding Hamiltonian)

9. Strong MF regime: 1) do eigenstates really overlap in space? Naïve expectation: A consequence of the dynamical scaling: k k weak space correlations strong space correlations - sparse fractals Numerical evidence:IQH WF: Chalker, Daniel (1988), Huckestein, Schweitzer (1994), Prack, Janssen, Freche (1996)Anderson transition in 3d: Brandes, Huckestein, Schweitzer (1996)WF of critical RMTs: Cuevas, Kravtsov (2007)

10. Strong MF regime: 1) do eigenstates really overlap in space? – averaged return probability for a wave packet P Expected scaling properties of P(t) 1 - dynamical scaling N - spatial scaling (IPR) - IR cutoff of the theory First analytical proof for the critical PLBRMT:(V.E. Kravtsov, A. Ossipov, O.Ye., 2010-2011) - strong MF Analytical results:

11. Strong MF regime: 2) are MF correlations phase independent? Retarded propagator of a wave-packet (density-density correlation function): Diffusion propagator in disordered systems with extended states (“Diffuson”)

12. Strong MF regime: 2) are MF correlations phase independent? Retarded propagator of a wave-packet (density-density correlation function): Dynamical scaling hypothesis for generalized diffuson: Q – momentum,  - DoS The same scaling exponent is possible if phase correlations are noncritical

13. Our current project 1 - small band width → almost diagonal MF RMT, strong multifractality b/|i-j|<<1 We calculate for this model 1) to check the hypothesis of dynamical scaling, 2) to study phase correlations in different regimes. Almost diagonal critical PLBRMT from the GOE and GUE symmetry classes

14. Method: The virial expansion Gas of low density ρ Almost diagonal RM bΔ b1 ρ1 Δ 2-level interaction 2-particle collision 3-particle collision b2 ρ2 3-level interaction As an alternative to the σ-model, we use the virial expansion in the number of interacting energy levels. Note: a field theoretical machinery of the –model cannot be used in the case of the strong fractality VE allows one to expand correlations functions in powers of b<<1(O.Ye., V.E. Kravtsov, 2003-2005);

15. SuSy virial expansion m Hmn Hmn Hmn n m n m n Hybridization of localized stated Interaction of energy levels SuSy is used to average over disorder(O.Ye., Ossipov, Kronmüller, 2007-2009) Coupling of supermatrices Summation over all possible configurations SuSy breaking factor

16. Virial expansion for generalized diffusion propagator → critical MF scaling is expected in Note: Coordinate-time representation: contribution from thediagonal part of RMT contribution fromj independent supermatrices The probability conservation: the sum rule:

17. Leading term of VE for generalized diffusion propagator 2-matrix approximation: Propagators at fixed time(r’=0) Critical statesat strong multifractality Extendedstates Lévy flight Diffusion

18. Leading term of VE: MF scaling Using the sum rule: Regime of dynamical scaling: • First conclusions • We have found a signature of (strong) multifractality with the scaling exponent = d2 ; • Dynamical scaling hypothesis has been confirmed for the generalized diffuson; • Phases of wave-functions do not have critical correlations.

19. Leading term of VE: Lévy flights - perturbative in Beyond regime of MF scaling: slow decay of correlations because of similar (but not MF) correlations of amplitudes and phases of (almost) localized states Meaning of - Lévi flights (due to long-range hopping) (heavy tailed probability of long steps →power-low tails in the probablity distribution of random walks)

20. Beyond leading term of VE: expected log-correcition Regime of dynamical scaling: due to dynamical scaling Subleading terms of the VE are needed to check this guess

21. Beyond leading term of VE: expected log-correcition Regime of Lévy flights: correlated phases → uncorrelated phases (WD-RMT) due to decorrelated phases Note: we expect that both exponents are renormalized with increasing b

22. Subleading term of VE for generalized diffusion propagator obeys the sum rule: Regime of Lévy flights: Regime of dynamical scaling: No log-corrections from As we expected Result 3-matrix approximation (GUE):

23. Phase correlations: crossover “strong-weak” MF regimes Perturbative (smooth crossover) Lévy flights exist at any finite b Non-perturbative (Kosterlizt-Thouless transition) Lévy flights exist only at b< bc Two different scenarios of the crossover/transition No long-rangecorrelations

24. Can we expect Lévy flights in short-ranged critical models? Smooth crossover Critical RMT vs. Anderson model Strong multifractality The Lévy flights exist Weak multifractality Instability of high gradients operators (Kravtsov, Lerner, Yudson, 1989)- a precursor of the Lévy flights in AM to be studied Guess: the Lévy flights may exist in both regimes (strong- and weak- multifractaity) and all (long-ranged and short-ranged) critical models.

25. Conclusions and further plans • We have demonstrated validity of the dynamical scaling hypothesis for the retarded propagator of the critical almost diagonal RMT. • Multifractal correlations and Lévy flights co-exist in the case of critical RMT. • We expect that the Lévy flights exist in regimes of strong- and weak- multifractality in long- and short- ranged critical models. • We plan to support our hypothesis by performing numerics at intermediate- and -model calculations at large- band width.