Multifractality of random wavefunctions: recent progress

1. Multifractality of random wavefunctions: recent progress V.E.Kravtsov Abdus Salam ICTP

2. Anderson transition disorder L Extendedstates Critical states Localizedstates

3. Multifractal wave functions Map of the regions with amplitude larger than the chosen level L L

4. Multifractal metal Multifractal insulator Multifractal metal and insulator

5. Quantitative description: fractal dimensions and spectrum of multifractality Weight of the map where wavefunction amplitude |y| ~ L is by definition L -a 2 f(a) L L Saddle-point approximation -> Legendre transform

6. Weak and strong fractality 4D 3D 2+e 4D Dq= d – g q 3D metal 2+e Weak fractality

7. PDF of wave function amplitude For weak multifractality Log-normal distribution with the variance ~ ln L Altshuler, Kravtsov, Lerner, 1986

8. Symmetry relationship Statistics of large and small amplitudes are connected! Mirlin, Fyodorov, 2006 Gruzberg,Ludwig,Zirnbauer, 2011

9. Unexpected consequence Small q shows that the sparse fractal is different from localization by statistically significant minimal amplitude Small moments exaggerate small amplitudes For infinitely sparse fractal

10. Supplement Dominated by large amplitudes Dominated by small amplitudes

11. Critical RMT: large- and small- bandwidth cases Mirlin & Fyodorov, 1996 Kravtsov & Muttalib, 1997 Kravtsov & Tsvelik 2000 criticality fractality Eigenstates are multifractal at all values of b d_2/d 2+e 1 3D Anderson, O class 0.6 1/b Weak fractality Strong fractality

12. pbb =1.64 pbb=1.39 pbb=1.26

13. Sigma-model: Valid for b>>1 Q=ULU is a geometrically constrained supermatrix: Duality! Y- functional: The nonlinear sigma-model and the dual representation Convenient to expand in small b for strong multifractality

14. 2-particle collision 3-particle collision Virial expansion in the number of resonant states Gas of low density ρ Almost diagonal RM bΔ b1 ρ1 2-level interaction Δ ρ2 b2 3-level interaction

15. Virial expansion as re-summation O.M.Yevtushenko, A.Ossipov, V.E.Kravtsov 2003-2011 F2 F3 Term containing m+1 different matrices Q gives the m-th term of the virial expansion

16. Virial expansion of correlation functions At the Anderson transition in d –dimensional space Each term proportional to gives a result of interaction of m+1 resonant states Parameter b enters both as a parameter of expansion and as an energy scale -> Virial expansion is more than the locator expansion

17. Small amplitude 100% overlap Metal: Large amplitude but rare overlap Insulator: Two wavefunction correlation: ideal metal and insulator

18. Critical enhancement of wavefunction correlations Amplitude higher than in a metal but almost full overlap States rather remote (d<<\E-E’|<E0) in energy are strongly correlated

19. Another difference between sparse multifractal and insulator wave functions

20. New length scale l0, new energy scale E0=1/r l0 Multifractal metal: x> l 0 3 Wavefunction correlations in a normal and a multifractal metal Critical power law persists Normal metal: x< l 0

21. ??? Density-density correlation function D(r,t)

22. Return probability formultifractal wave functions Kravtsov, Cuevas, 2011 Numerical result Analytical result

23. Quantum diffusion at criticality and classical random walk on fractal manifolds Quantum critical case Random walks on fractals Similarity of description!

24. Oscillations in return probability Akkermans et al. EPL,2009 Classical random walk on regular fractals Multifractal wavefunctions Analytical result Kravtsov, Cuevas, 2011

25. Real experiments