V.E.Kravtsov ICTP, Trieste and Landau Institute Collaboration: Ivan Khaymovich, Aaalto Emilio Cuevas, Murcia

rr site disorder: branching number K Energy E = 0

Giulio Biroli, 2011: ET - YES De Luca et al. 2014: ET – No, only non- ergodic phase Mirlin & Fyodorov: : ET: No, only ergodic phase Extrapolation from N=2K- 32K

Structural disorder: WD (Uzy Smiliansky) On-site energy disorder: P  

GOE: Special diagonal: Rosenzweig- Porter ensemble Adding diagonal disorder

Two-level correlation function (a generalization of Kunz and Shapiro) R(    R(    Level compressibility 

GUE Diagonal RM Izikson-Zuber formula of integrating over unitary matrices diagonalizing H

Rescaling and N-> infinity Depends on the distribution function of the diagonal matrix A=diag{a}

tend to a finite limit independent of  cancels out  Poisson due to infinitely fast oscillations  regular oscillations 1<  no oscillations  jump to the GUE form  

 1 2 localized Extended, non-ergodic Extended ergodic

1/lnN

There is some curvature!

ideal insulator GOE

Rosenzweig-Porter two transition points

only one transition pointy K=2

 Rosenzweig-Porter RM model contains both Anderson and Ergodic transitions  They are seen in the rigorous theory of the two-level correlation function  Perturbative treatment of the eigenfunction statistics gives the three phases: localized, non-ergodic extended, ergodic extended  Numerics confirm existence of three phases  moments (Shannon entropy) vs lnN has a curvature which changes sign at the transitions.