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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.: A A A A A A A A A A. Collaborators:. References:.
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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
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
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)
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
Fractal dimensions Moments: Extended states: Localized states: Anomalous scaling exponents: Critical point: How one can calculate ? Green’s functions:
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)
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)
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
Strong multifractality in the ultrametric ensemble General expression: Ultrametric random matrices: Fractal dimensions: Y. V. Fyodorov, AO, A. Rodriguez, J. Stat. Mech., L12001 (2009)
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)
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
Fractal dimensions: beyond universality can be choosen the same for all models model specific Can we calculate ?
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)
Weak multifractality How one can calculate ? Mapping onto the non-linear σ-model Perturbative expansion in the regime
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)
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
2D power-law random hopping model Strong criticality critical at CFT prediction: AO, I. Rushkin, E. Cuevas, arXiv:1101.2641
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
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
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)
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)
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
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)
2D power-law random hopping model AO, I. Rushkin, E. Cuevas, arXiv:1101.2641