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

Anderson transition disorder L Extended states Critical states Localized states

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

Multifractal metal and insulator Multifractal insulator

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

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

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

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

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

Supplement Dominated by large amplitudes Dominated by small amplitudes

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

pbb =1.64 pbb=1.39 pbb=1.26

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

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

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

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

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

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

Another difference between sparse multifractal and insulator wave functions

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

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

Return probability for multifractal wave functions Kravtsov, Cuevas, 2011 Numerical result Analytical result

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

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

Real experiments