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

Alexander Ossipov Alexander Ossipov

School of Mathematical Sciences, University of Nottingham, UK

1

Critical eigenstates of theCritical eigenstates of the

long-range random Hamiltonianslong-range random Hamiltonians

2

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)

3

4

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:

5

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

6

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:

7

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.

8

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

11

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

12

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

14

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

?

16

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

17

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:

18

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:

19

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!

20

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

22

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)

23

2D power-law random hopping model 2D power-law random hopping model

AO, I. Rushkin, E. Cuevas, arXiv:1101.264124