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

2. Outline of the talk  Unconventional RMT with fractal eigenstates  Multifractality and Scaling properties of correlation functions 1.Introduction:  Unconventional RMT with fractal eigenstates  Multifractality and Scaling properties of correlation functions 2.Strong multifractality regime:  Basic ideas of SuSy Virial Expansion  Application of SuSyVE for generalized diffusion propagator  Results and Discussion: multifractality, Lévy flights, phase correlations 3.Scaling of retarded propagator:  Results and Discussion: multifractality, Lévy flights, phase correlations 4.Conclusions and open questions Bielefeld, 16 December 2011

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

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

5. MF RMT: Power-Law-Banded Random Matrices 2 π b>>1 weak multifractality 1-d 2 <<1 – regime of weak multifractality b<<1 strong multifractality d 2 <<1 – regime of strong multifractality b is the bandwidth b RMT with multifractal eignestates at any band-width 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 If ω>  then must play a role of L: L  LωLω ω For a disordered system at the critical point (fractal eigenstates) Two point correlation function : Critical correlations (Wegner, 1985; Chalker, Daniel, 1988; Chalker 1980)

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

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

9. k Naïve expectation: weak space correlations Strong MF regime: 1) do eigenstates really overlap in space? - sparse fractals k A consequence of the dynamical scaling: strong space correlations 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? First analytical proof for the critical PLBRMT: (V.E. Kravtsov, A. Ossipov, O.Ye., 2010-2011) - strong MF – averaged return probability for a wave packet - spatial scaling (IPR) 1 N - dynamical scaling Expected scaling properties of P(t) P - IR cutoff of the theory 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: The same scaling exponent is possible if phase correlations are noncritical Q – momentum,  - DoS

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

14. σ-model the virial expansion in the number of interacting energy levels As an alternative to the σ-model, we use the virial expansion in the number of interacting energy levels. Method: The virial expansion 2-particle collision Gas of low density ρ 3-particle collision ρ1ρ1 ρ2ρ2 Almost diagonal RM b1b1 2-level interaction Δ bΔbΔ b2b2 3-level interaction VE allows one to expand correlations functions in powers of b<<1 (O.Ye., V.E. Kravtsov, 2003-2005); Note: a field theoretical machinery of the  –model cannot be used in the case of the strong fractality

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

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

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

18. Leading term of VE: MF scaling Regime of dynamical scaling: Using the sum rule: First conclusions We have found a signature of (strong) multifractality with the scaling exponent = d 2 ; 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 Beyond regime of MF scaling: 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) slow decay of correlations because of similar (but not MF) correlations of amplitudes and phases of (almost) localized states - perturbative in

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 Note: we expect that both exponents are renormalized with increasing b correlated phases → uncorrelated phases (WD-RMT) Regime of Lévy flights: due to decorrelated phases

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

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

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

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.