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

2. Anderson transition disorder L Extended states Critical states
Localized states

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

4. Multifractal metal and insulator
Multifractal insulator

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

6. Weak and strong fractality
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. Eigenstates are multifractal at all values of b
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. The nonlinear sigma-model and the dual representation
Valid for b>>1
Duality!
Q=ULU is a geometrically constrained supermatrix:
Y- functional:
Convenient to expand in small b for strong multifractality

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

15. Virial expansion as re-summation
O.M.Yevtushenko, A.Ossipov, V.E.Kravtsov 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. Two wavefunction correlation: ideal metal and insulator
Small amplitude 100% overlap
Insulator:
Large amplitude but rare overlap

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

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

22. Return probability for multifractal 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