Oleg Yevtushenko

Critical scaling in Random Matriceswith fractal eigenstates

In collaboration with: Vladimir Kravtsov (ICTP, Trieste), Alexander Ossipov (University of Nottingham)Emilio Cuevas (University of Murcia)

Ludwig-Maximilians-Universität, München, GermanyArnold Sommerfeld Center for Theoretical Physics

Brunel, 18 December 2009

Outline of the talkOutline of the talk

1. Introduction: Unconventional RMT with fractal eigenstates Unconventional RMT with fractal eigenstates Local Spectral correlation function: Scaling properties Local Spectral correlation function: Scaling properties

2. Strong multifractality regime: Virial Expansion: basic ideas Virial Expansion: basic ideas Application of VE for critical exponentsApplication of VE for critical exponents

3. Scaling exponents: Calculations: Contributions of 2 and 3 overlapping eigenstates Calculations: Contributions of 2 and 3 overlapping eigenstates Speculations: Scenario for universality and Duality Speculations: Scenario for universality and Duality

4. Conclusions

Brunel, 18 December 2009

WD and Unconventional Gaussian RMT

22 10, , ( )ij ii i jH H H F i j

Statistics of RM entries:

Parameter β reflects symmetry classes (β=1: GOE, β=2: GUE)

If F(i-j)=1/2 → the Wigner–Dyson (conventional) RMT (1958,1962):

Factor A parameterizes spectral statistics and statistics of eigenstates

F(x)

x1

A

Function F(i-j) can yield universality classes of the eigenstates, universality classes of the eigenstates, different from WD RMTdifferent from WD RMT

Generic unconventional RMT:

H - Hermithian matrix with random (independent, Gaussian-distributed) entries

†ˆ ˆ ˆ,n n nH H H The Schrödinger equation for a 1d chain:(eigenvalue/eigenvector problem)

3 cases which are important for physical applications

4

12

,n n

n k

k E PInverse Participation Ratio

2

1dN N

2P 20 d d (fractal) dimension of a support:(fractal) dimension of a support:the space dimension d=1 for RMT

k

2

n k

n

m

extended extended (WD)

Model for metalsModel for metals

2 1d

k

2

n k fractalfractal

nm

Model for systemsModel for systemsat the critical pointat the critical point

20 1d

k

2

n k

n m

localizedlocalized

Model for insulatorsModel for insulators

2 0d

MF RMT: Power-Law-Banded Random Matrices

2

, 2

1

1i jH

i jb

2 π b>>1

2const1 2d b

1-d2<<1 – regime of weak multifractalityweak multifractality

b<<1

2 const d b

d2<<1 – regime of strong multifractalitystrong multifractality

2

2,| | ~

1 ,

1,

i jH

i j

i j b

i j b

b is the bandwidth

b

RMT with multifractal eignestates at any band-widthRMT with multifractal eignestates at any band-width

(Mirlin, Fyodorov et.al., 1996, Mirlin, Evers, 2000 - GOE and GUE symmetry classes)

Correlations of MF eigenstates

Local in space two point spectral correlation function (LDOS-LDOS):

d –space dimension, L – system size, - mean level spacing

For a disordered system at the critical point (MF eigenstates)

(Wegner, 1985)

If ω> then must play a role of L:

A dynamical scaling assumption:

(Chalker, Daniel, 1988; Chalker, 1990)

MF enhancement of eigenstate correlations: the Anderson model

The Anderson model: tight binding Hamiltonian of a disordered system (3-diagonal RM)

The Chalkers’ scaling:

(Cuevas , Kravtsov, 2007)

extended

localizedcritical

Extended states: small amplitudehigh probability of overlap in space

Localized states: high amplitudesmall probability of overlap in space

the fractal eigenstates the fractal eigenstates strongly overlap in spacestrongly overlap in space

MF states: relatively high amplitude and

- Enhancement of correlations

Universality of critical correlations: MF RMT vs. the Anderson model

MF (critical) RMT, bandwidth b

Anderson model at criticality (MF eigenstates),dimension d

(Cuevas, Kravtsov, 2007)

“” – MF PLBRMT, β=1, b = 0.42

“” – 3d Anderson model from orthogonal class with MF eigenstates at the mobility edge, E=3.5

Advantages of the critical RMT: 1) numerics are not very time-consuming; 2) it is known how to apply the SuSy field theory

k

2

n k

nm

Naïve expectation:

weak space correlations

Strong MF regime: do eigenstates really overlap in space?

- sparse fractals

k

2

n k

nm

A consequence of the Chalker’s scaling:

strong space correlations

So far, no analytical check of the Chalker’s scaling; just a numerical evidence

Our goal: we study the Chalker’s ansatz for the scaling relation in the strong multifractality regime using the model of the MF RMT with a small bandwidth

2

222

1 1 1 1, , 1

2 2 21

ii i j

bH H b

i ji jb

Almost diagonal RMT from the GUE symmetry class

Statement of problem

Method: The virial expansion

2-particle collision

Gas of low density ρ

3-particle collision

ρ1

ρ2

Almost diagonal RM

b1

2-level interaction

b

Δ

bΔ

b2

3-level interaction

VE for RMT: 1) the Trotter formula & combinatorial analysis (OY, Kravtsov, 2003-2005); 2) Supersymmetric FTSupersymmetric FT (OY, Ossipov, Kronmueller, 2007-2009).

VE allows one to expand correlation functions in powers of b<<1

Correlation function and expected scaling in time domain

It is more convenient to use the VE in a time domain

– return probability for a wave packet

VE for the return probability:VE for the return probability:

ExpectedExpected scaling properties scaling properties

( - scaled time)

- the IPR spatial scaling

- the Chalker’s dynamical scaling

O(b1) O(b2)

VE for the scaling exponentVE for the scaling exponent

What shall we calculate and check?

2 level contribution of the VE 3 level contribution of the VE

1) Log-behavior:

2) The scaling assumption:

are constants

log2(…) must cancel out in P(3)- (P(2))2/2

3) The Chalker’s relation for exponents =1-d2 (z=1)

Part I: Calculations

Part II: Scenarios and speculations

Regularization of logarithmic integrals

- are sensitive to small distances

Assumption:Assumption: the scaling exponents are not sensitive to small distances

Discrete system: summation over 1d lattice in all terms of the VE

small distances are regularized by the lattice

More convenient regularization:

small distances are regularized by the variance

VE for the return probability

Two level contribution

Three level contribution

- is known but rather cumbersome (details of calculations can be discussed after the talk)

Two level contribution

The scaling assumption and the relation The scaling assumption and the relation =1-d=1-d22 hold true up to hold true up to O(b)O(b)

HHomogeneity of the argumentomogeneity of the argument at at → 0→ 0

The leading term of the virial expansion

Three level contribution

The scaling assumption holds true up to the terms of order The scaling assumption holds true up to the terms of order O(bO(b2 2 loglog22(())))

homogeneous arguments β/x and β/y at →0

The subleading term of the virial expansion

Calculations: cancel out in P(3)- (P(2))2/2

Part I: Calculations

Part II: Scenarios and speculations

Is the Chalker’s relation =1-d2 exact?

Regime of strong multifractality

(Cuevas, O.Ye., unpublished)

(Cuevas, Kravtsov, 2007)

Which conditions (apart from the homogeneity property) arenecessary to prove universality of subleading terms of order O( b2 ) in the scaling exponents?

Intermediate regime

Numerics confirm that the Chalker’srelation is exact and holds true for any b.

Integral representations:

Universality of the scaling exponent

The homogeneity property results in:

Assumption: the scaling exponents do not contain anomalous contributions (coming from uncertainties )

then

The Chalker’s relation The Chalker’s relation =1-d=1-d22 holds true up to holds true up to O(bO(b22) ) (a hint that it is exact)

Sub-leading contributions to the scaling exponents:

Duality of scaling exponents: small vs. large b-parameter

(Kravtsov, arXiv:0911.06, Kravtsov, Cuevas, O.Ye., Ossipov [in progress] )

Note that at B<<1 & at B >>1

Does this equality hold true only at small-/large- or at arbitrary B?Does this equality hold true only at small-/large- or at arbitrary B?

Yes – it holds true for arbitrary B!Yes – it holds true for arbitrary B!

If it is the exact relation between d2(B) and d2(1/B) →

Duality between the regimes of Duality between the regimes of strong and weak multifractality!?strong and weak multifractality!?

- Strong multifractality (b << 1) - Weak multifractality (b >> 1)

Conclusions and open questions

• We have studied critical dynamical scaling using the model of the of the almost diagonal RMT with multifractal eigenstates

• We have proven that the Chalker’s scaling assumption holds trueup to the terms of order O(bO(b2 2 loglog22(())))

• We have proven that the Chalker’s relation =1-d=1-d22 holds true up to the terms of order O(b) O(b)

• We have suggested a schenario which (under certain assumptions) expains why the Chalker’s relation =1-d=1-d22 holds true up to the terms of order O(bO(b22) ) –a hint that the Chalker’s relation is exact

• We plan a) to generalize the results accounting for an interaction

of arbitrary number of levelsb) to study duality in the RMT with multifractal eigenstates

j j jQ

commuting variablesanticommuting variables

Supermatrix:

/ 1ˆ ( )ˆ 0

R AG EE H i

- Retarded/Advanced Green’s functions (resolvents):

The supersymmetric action for RMT

One-matrix part of action

2

(1, 1)

1

2

RA

aa

diag

a H

Weak “interaction” of supermatrices

breaks SuSy in R/A sectors

SuSy virial expansion for almost diagonal RMs

Perturbation theory in off-diagonal matrix elements

Interaction of 2 matrices(of 2 localized eigenstates)Qm Qn…

m n

2

mnH(2)V

Subleading terms: Interaction of 3, 4 … matrices(3,4, )V

etc.(Mayer’s function)Qm Qn…

m n

Localized eigenstates →noninteracting Q-matrices

Diagonal part of RMT

DV

Let’s rearrange“interacting part”

* *

/

*

exp

exp

i jR Aij

d d iSG

d d iS

The problem of denominator: How to average over disorder?

jijijij i HES )( / ( , ) * exp ( ) ( )R A

ij i jG E d d iS iS

)(exp

)(exp

1 *

*

1

iSdd

iSddZThe supersymmetry

trick

/ 1ˆ ( )ˆ 0

R AG EE H i

- Retarded/Advanced Green’s functions (resolvents):

Method: Green’s functions and SuSy representation

( ) ( ) ; ( ) ( )i ij ij j i ij ij jij ijS E H S E H