1. Alexander Ossipov School of Mathematical Sciences, University of Nottingham, UK TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AAA A A A AA A A 1 Critical eigenstates of the long-range random Hamiltonians

2. TexPint fonts used in EMF. Read the TexPoint manual before you delete this box.: AAA A A A AA A A 2 Collaborators: Yan Fyodorov, Ilia Rushkin Vladimir Kravtsov Oleg Yevtushenko Emilio Cuevas Alberto Rodriguez References: J. Stat. Mech., L12001 (2009) PRB 82, 161102(R) (2010) J. Stat. Mech. L03001 (2011) J. Phys. A 44, 305003 (2011) arXiv:1101.2641

3. Anderson model Hamiltonian on a d-dimensional lattice: Metal-insulator transition in the three-dimensional case: W W c ergodic(multi)fractallocalized Banded RM Wigner-Dyson RMPower-law Banded RM P. W. Anderson, Phys. Rev. 109, 1492 (1958) 3

4. 4 Outline 1.Power-law banded matrices and ultrametric ensemble: universality of the fractal dimensions ensemble: universality of the fractal dimensions 2. Fractal dimensions: beyond the universality 3. Criticality in 2D long-range hopping model 4. Dynamical scaling: validity of Chalker’s ansatz

5. Fractal dimensions Extended states: Localized states: Anomalous scaling exponents: Critical point: How one can calculate ? Green’s functions: 5 Moments:

6. Power-law banded random matrices mapping onto the non-linear σ-model weak multifractality almost diagonal matrix strong multifractality A. D. Mirlin et. al., Phys. Rev. E 54, 3221 (1996) Gaussian distributed, independent critical states at all values of 6

7. Ultrametricensemble Ultrametric ensemble Random hopping between boundary nodes of a tree of K generations with coordination number 2 Y. V. Fyodorov, AO, A. Rodriguez, J. Stat. Mech., L12001 (2009) Distance number of edges in the shortest path connecting i and j --- ultrametric Strong triangle inequality: 7

8. Almost diagonal matrices localized states extended states critical states determines the nataure of eigenstates in the thermodynamic limit If, then the moments can be calculated perturbatevely. 8

9. Strong multifractality in the ultrametric ensemble the ultrametric ensemble Ultrametric random matrices: General expression: Fractal dimensions: Y. V. Fyodorov, AO, A. Rodriguez, J. Stat. Mech., L12001 (2009) 9

10. Universality of fractal dimensions Power-law banded matrices: Ultrametric random matrices: universality A. D. Mirlin and F. Evers, Phys. Rev. B 62, 7920 (2000) Y. V. Fyodorov, AO, A. Rodriguez, J. Stat. Mech., L12001 (2009) 10

11. 11 Outline 1.Power-law banded matrices and ultrametric ensemble: universality of the fractal dimensions ensemble: universality of the fractal dimensions 2. Fractal dimensions: beyond the universality 3. Criticality in 2D long-range hopping model 4. Dynamical scaling: validity of Chalker’s ansatz

12. Fractal dimensions: beyond universality can be choosen the same for all models Can we calculate ? model specific 12

13. Fractal dimension d 2 for power-law banded matrices Supersymmetric virial expansion: where V. E. Kravtsov, AO, O. M. Yevtushenko, E. Cuevas, Phys. Rev. B 82, 161102(R) (2010) V. E. Kravtsov, AO, O. M. Yevtushenko, J. Phys. A 44, 305003 (2011) 13

14. Weak multifractality Mapping onto the non-linear σ-model How one can calculate ? Perturbative expansion in the regime 14

15. Non-universal contributions to the fractal dimensions Anomalous fractal dimensions models are different Ultrametric ensemble Power-law ensemble I. Rushkin, AO, Y. V. Fyodorov, J. Stat. Mech. L03001 (2011) 15 couplings of the sigma-models ?

16. 16 Outline 1.Power-law banded matrices and ultrametric ensemble: universality of the fractal dimensions ensemble: universality of the fractal dimensions 2. Fractal dimensions: beyond the universality 3. Criticality in 2D long-range hopping model 4. Dynamical scaling: validity of Chalker’s ansatz

17. 17 2D power-law random hopping model critical at Strong criticality AO, I. Rushkin, E. Cuevas, arXiv:1101.2641 CFT prediction:

18. 18 Weak criticality 2D power-law random hopping model AO, I. Rushkin, E. Cuevas, arXiv:1101.2641 Perturbative calculations in the non-linear σ-model: Propagator Non-fractal wavefunctions:

19. 19 Outline 1.Power-law banded matrices and ultrametric ensemble: universality of the fractal dimensions ensemble: universality of the fractal dimensions 2. Fractal dimensions: beyond the universality 3. Criticality in 2D long-range hopping model 4. Dynamical scaling: validity of Chalker’s ansatz

20. Spectral correlations J.T. Chalker and G.J.Daniell, Phys. Rev. Lett. 61, 593 (1988) Strong multifractality: Strong overlap of two infinitely sparse fractal wave functions! 20

21. Return probability Strong multifractality : V. E. Kravtsov, AO, O. M. Yevtushenko, E. Cuevas, Phys. Rev. B 82, 161102(R) (2010) V. E. Kravtsov, AO, O. M. Yevtushenko, J. Phys. A 44, 305003 (2011) 21

22. 22 Summary Two critical random matrix ensembles: the power-law randomTwo critical random matrix ensembles: the power-law random matrix model and the ultrametric model matrix model and the ultrametric model Analytical results for the multifractal dimensions in the regimesAnalytical results for the multifractal dimensions in the regimes of the strong and the weak multifractality of the strong and the weak multifractality Universal and non-universal contributions to the fractal dimensionsUniversal and non-universal contributions to the fractal dimensions Non-fractal wavefunctions in 2D critical random matrix ensembleNon-fractal wavefunctions in 2D critical random matrix ensemble Equivalence of the spectral and the spatial scaling exponentsEquivalence of the spectral and the spatial scaling exponents

23. Fractal dimensions in the ultrametric ensemble Y. V. Fyodorov, AO and A. Rodriguez, J. Stat. Mech., L12001 (2009) Anomalous exponents: Symmetry relation: A. D. Mirlin et. al., Phys. Rev. Lett. 97, 046803 (2007) 23

24. 2D power-law random hopping model 2D power-law random hopping model AO, I. Rushkin, E. Cuevas, arXiv:1101.2641 24