alexander ossipov school of mathematical sciences, university of nottingham, uk
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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:. - PowerPoint PPT PresentationTRANSCRIPT
Alexander Ossipov Alexander Ossipov
School of Mathematical Sciences, University of Nottingham, UK
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Critical eigenstates of theCritical eigenstates of the
long-range random Hamiltonianslong-range random Hamiltonians
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Collaborators:Collaborators:
Yan Fyodorov, Ilia RushkinYan Fyodorov, Ilia Rushkin
Vladimir KravtsovVladimir Kravtsov
Oleg YevtushenkoOleg Yevtushenko
Emilio Cuevas Emilio Cuevas
Alberto RodriguezAlberto Rodriguez
References: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
Anderson modelAnderson modelHamiltonian on a d-dimensional lattice:
Metal-insulator transition in the three-dimensional case:
W<Wc W=Wc W>Wc
ergodic (multi)fractal localized
Banded RMWigner-Dyson RM Power-law Banded RMP. W. Anderson, Phys. Rev. 109, 1492 (1958)
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OutlineOutline
1.1. Power-law banded matrices and ultrametric Power-law banded matrices and ultrametric ensemble: universality of the fractal dimensionsensemble: universality of the fractal dimensions
2. Fractal dimensions: beyond the universality 2. Fractal dimensions: beyond the universality
3. Criticality in 2D long-range hopping model3. Criticality in 2D long-range hopping model
4. Dynamical scaling: validity of Chalker’s ansatz4. Dynamical scaling: validity of Chalker’s ansatz
Fractal dimensions Fractal dimensions
I q =X
r
jÃn(r)j2q
®/ L ¡ dq(q¡ 1)
Extended states: Localized states:
Anomalous scaling exponents:
Critical point:
How one can calculate ?
Green’s functions:
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Moments:
Power-law banded random Power-law banded random matricesmatrices
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
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UltrametricUltrametric ensembleensemble
Random hopping between boundary nodesof a tree of K generations with coordinationnumber 2
Y. V. Fyodorov, AO, A. Rodriguez, J. Stat. Mech., L12001 (2009)
Distance number of edges in the shortestpath connecting i and j --- ultrametric
Strong triangle inequality:
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Almost diagonal matricesAlmost 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.
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Strong multifractality inStrong multifractality in the ultrametric ensemblethe ultrametric ensemble
Ultrametric random matrices:
General expression:
Fractal dimensions:
Y. V. Fyodorov, AO, A. Rodriguez, J. Stat. Mech., L12001 (2009)9
Universality of fractal dimensionsUniversality 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
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OutlineOutline
1.1. Power-law banded matrices and ultrametric Power-law banded matrices and ultrametric ensemble: universality of the fractal dimensionsensemble: universality of the fractal dimensions
2. Fractal dimensions: beyond the universality 2. Fractal dimensions: beyond the universality
3. Criticality in 2D long-range hopping model3. Criticality in 2D long-range hopping model
4. Dynamical scaling: validity of Chalker’s ansatz4. Dynamical scaling: validity of Chalker’s ansatz
Fractal dimensions: Fractal dimensions: beyond universalitybeyond universality
can be choosen the same for all models
Can we calculate ?
model specific
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Fractal dimension Fractal dimension dd22 for power-lawfor power-law
banded matrices 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
Weak multifractality Weak multifractality
Mapping onto the non-linear σ-model
How one can calculate ?
Perturbative expansion in the regime
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Non-universal contributions to Non-universal contributions to the fractal dimensions 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
?
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OutlineOutline
1.1. Power-law banded matrices and ultrametric Power-law banded matrices and ultrametric ensemble: universality of the fractal dimensionsensemble: universality of the fractal dimensions
2. Fractal dimensions: beyond the universality 2. Fractal dimensions: beyond the universality
3. Criticality in 2D long-range hopping model3. Criticality in 2D long-range hopping model
4. Dynamical scaling: validity of Chalker’s ansatz4. Dynamical scaling: validity of Chalker’s ansatz
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2D power-law random hopping model 2D power-law random hopping model
critical at
Strong criticality Strong criticality
AO, I. Rushkin, E. Cuevas, arXiv:1101.2641
CFT prediction:
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Weak criticality Weak criticality
2D power-law random hopping model 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:
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OutlineOutline
1.1. Power-law banded matrices and ultrametric Power-law banded matrices and ultrametric ensemble: universality of the fractal dimensionsensemble: universality of the fractal dimensions
2. Fractal dimensions: beyond the universality 2. Fractal dimensions: beyond the universality
3. Criticality in 2D long-range hopping model3. Criticality in 2D long-range hopping model
4. Dynamical scaling: validity of Chalker’s ansatz4. Dynamical scaling: validity of Chalker’s ansatz
Spectral correlationsSpectral 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!
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Return probabilityReturn 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
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SummarySummary
• Two critical random matrix ensembles: the power-law random Two critical random matrix ensembles: the power-law random matrix model and the ultrametric modelmatrix model and the ultrametric model
• Analytical results for the multifractal dimensions in the regimes Analytical results for the multifractal dimensions in the regimes of the strong and the weak multifractalityof 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
Fractal dimensions in Fractal dimensions in the ultrametric ensemblethe 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)
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2D power-law random hopping model 2D power-law random hopping model
AO, I. Rushkin, E. Cuevas, arXiv:1101.264124