1. Oleg Yevtushenko Critical scaling in Random Matrices with fractal eigenstates In collaboration with: Vladimir Kravtsov (ICTP, Trieste), Alexander Ossipov (University of Nottingham) Emilio Cuevas (University of Murcia) Ludwig-Maximilians-Universität, München, Germany Arnold Sommerfeld Center for Theoretical Physics Brunel, 18 December 2009

2. Outline of the talk  Unconventional RMT with fractal eigenstates  Local Spectral correlation function: Scaling properties 1.Introduction:  Unconventional RMT with fractal eigenstates  Local Spectral correlation function: Scaling properties 2.Strong multifractality regime:  Virial Expansion: basic ideas  Application of VE for critical exponents  Calculations: Contributions of 2 and 3 overlapping eigenstates  Speculations: Scenario for universality and Duality 3.Scaling exponents:  Calculations: Contributions of 2 and 3 overlapping eigenstates  Speculations: Scenario for universality and Duality 4.Conclusions Brunel, 18 December 2009

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 fractal Model for systems at the critical point k localized Model for insulators

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 MF eigenstates Local in space two point spectral correlation function (LDOS-LDOS): d –space dimension, L – system size,  - mean level spacing For a disordered system at the critical point (MF eigenstates) (Wegner, 1985) If ω>  then must play a role of L: A dynamical scaling assumption: (Chalker, Daniel, 1988; Chalker, 1990)

7. MF enhancement of eigenstate correlations: the Anderson model The Anderson model: tight binding Hamiltonian of a disordered system (3-diagonal RM) The Chalkers’ scaling: (Cuevas, Kravtsov, 2007) extended localized critical Extended states: small amplitude high probability of overlap in space Localized states: high amplitude small probability of overlap in space the fractal eigenstates strongly overlap in space MF states: relatively high amplitude and - Enhancement of correlations

8. Universality of critical correlations: MF RMT vs. the Anderson model MF (critical) RMT, bandwidth b Anderson model at criticality (MF eigenstates), dimension d (Cuevas, Kravtsov, 2007) “  ” – MF PLBRMT, β=1, b = 0.42 “  ” – 3d Anderson model from orthogonal class with MF eigenstates at the mobility edge, E=3.5 Advantages of the critical RMT: 1) numerics are not very time-consuming; 2) it is known how to apply the SuSy field theory

9. k Naïve expectation: weak space correlations Strong MF regime: do eigenstates really overlap in space? - sparse fractals k A consequence of the Chalker’s scaling: strong space correlations So far, no analytical check of the Chalker’s scaling; just a numerical evidence

10. Our goal:we study the Chalker’s ansatz for the scaling relation in the strong multifractality regime using the model of the MF RMT with a small bandwidth Almost diagonal RMT from the GUE symmetry class Statement of problem

11. 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 Supersymmetric FT VE for RMT: 1) the Trotter formula & combinatorial analysis (OY, Kravtsov, 2003-2005); 2) Supersymmetric FT (OY, Ossipov, Kronmueller, 2007-2009). VE allows one to expand correlation functions in powers of b<<1

12. Correlation function and expected scaling in time domain It is more convenient to use the VE in a time domain – return probability for a wave packet VE for the return probability: Expected scaling properties (  - scaled time) - the IPR spatial scaling - the Chalker’s dynamical scaling O(b 1 )O(b 2 ) VE for the scaling exponent

13. What shall we calculate and check? 2 level contribution of the VE3 level contribution of the VE 1) Log-behavior: 2) The scaling assumption: are constants log 2 (…) must cancel out in P (3) - (P (2) ) 2 /2 3) The Chalker’s relation for exponents  =1-d 2 (z=1)

14. Part I: Calculations Part II: Scenarios and speculations

15. Regularization of logarithmic integrals - are sensitive to small distances Assumption: Assumption: the scaling exponents are not sensitive to small distances Discrete system: summation over 1d lattice in all terms of the VE small distances are regularized by the lattice More convenient regularization: small distances are regularized by the variance

16. VE for the return probability Two level contribution Three level contribution - is known but rather cumbersome (details of calculations can be discussed after the talk)

17. Two level contributionThe scaling assumption and the relation =1-d2 hold true up to O(b) Homogeneity of the argument at  → 0 The leading term of the virial expansion

18. Three level contributionThe scaling assumption holds true up to the terms of order O(b2 log2()) homogeneous arguments β/x and β/y at  →0 The subleading term of the virial expansion Calculations: cancel out in P (3) - (P (2) ) 2 /2

19. Part I: Calculations Part II: Scenarios and speculations

20. Is the Chalker’s relation  =1-d 2 exact? Regime of strong multifractality (Cuevas, O.Ye., unpublished) (Cuevas, Kravtsov, 2007) Which conditions (apart from the homogeneity property) are necessary to prove universality of subleading terms of order O( b 2 ) in the scaling exponents? Intermediate regime Numerics confirm that the Chalker’s relation is exact and holds true for any b.

21. Integral representations: Universality of the scaling exponent The homogeneity property results in: Assumption: the scaling exponents do not contain anomalous contributions (coming from uncertainties ) thenThe Chalker’s relation =1-d2 holds true up to O(b2) (a hint that it is exact) Sub-leading contributions to the scaling exponents:

22. Duality of scaling exponents: small vs. large b-parameter (Kravtsov, arXiv:0911.06, Kravtsov, Cuevas, O.Ye., Ossipov [in progress] ) Note that at B >1Does this equality hold true only at small-/large- or at arbitrary B? Yes – it holds true for arbitrary B! Duality between the regimes of strong and weak multifractality!? Yes – it holds true for arbitrary B! If it is the exact relation between d 2 (B) and d 2 (1/B) → Duality between the regimes of strong and weak multifractality!? - Strong multifractality (b > 1)

23. Conclusions and open questions We have studied critical dynamical scaling using the model of the of the almost diagonal RMT with multifractal eigenstates O(b 2 log 2 (  ))We have proven that the Chalker’s scaling assumption holds true up to the terms of order O(b 2 log 2 (  ))  =1-d 2 O(b)We have proven that the Chalker’s relation  =1-d 2 holds true up to the terms of order O(b)  =1-d 2 O(b 2 )We have suggested a schenario which (under certain assumptions) expains why the Chalker’s relation  =1-d 2 holds true up to the terms of order O(b 2 ) – a hint that the Chalker’s relation is exact We plan a) to generalize the results accounting for an interaction of arbitrary number of levels b) to study duality in the RMT with multifractal eigenstates

26. commuting variables anticommuting variables Supermatrix: - Retarded/Advanced Green’s functions (resolvents): The supersymmetric action for RMT One-matrix part of action Weak “interaction” of supermatrices breaks SuSy in R/A sectors

27. SuSy virial expansion for almost diagonal RMs Perturbation theory in off-diagonal matrix elements Interaction of 2 matrices (of 2 localized eigenstates) QmQm QnQn … mn Subleading terms: Interaction of 3, 4 … matrices etc. (Mayer’s function) QmQm QnQn … mn Localized eigenstates → noninteracting Q-matrices Diagonal part of RMT Let’s rearrange “interacting part”

29. The problem of denominator: How to average over disorder? The supersymmetry trick - Retarded/Advanced Green’s functions (resolvents): Method: Green’s functions and SuSy representation