Alexander Ossipov School of Mathematical Sciences, University of Nottingham, UK

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

2. Collaborators: 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 Yan Fyodorov, Ilia Rushkin Vladimir Kravtsov Oleg Yevtushenko Emilio Cuevas Alberto Rodriguez TexPint fonts used in EMF. Read the TexPoint manual before you delete this box.: AAAAAAAAAA

3. ergodic (multi)fractal localized Anderson model Hamiltonian on a d-dimensional lattice: Metal-insulator transition in the three-dimensional case: W<Wc W=Wc W>Wc Banded RM Wigner-Dyson RM Power-law Banded RM P. W. Anderson, Phys. Rev. 109, 1492 (1958)

4. Outline • Power-law banded matrices and ultrametric • 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 Moments: Extended states: Localized states: Anomalous scaling exponents: Critical point: How one can calculate ? Green’s functions:

6. Power-law banded random matrices Gaussian distributed, independent critical states at all values of 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)

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

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

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

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

11. Outline • Power-law banded matrices and ultrametric • 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 model specific Can we calculate ?

13. Fractal dimension d2for 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)

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

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

16. Outline • Power-law banded matrices and ultrametric • 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. 2D power-law random hopping model Strong criticality critical at CFT prediction: AO, I. Rushkin, E. Cuevas, arXiv:1101.2641

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

19. Outline • Power-law banded matrices and ultrametric • 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 Strong multifractality: Strong overlap of two infinitely sparse fractal wave functions! J.T. Chalker and G.J.Daniell, Phys. Rev. Lett. 61, 593 (1988)

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)

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

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

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