Oleg Yevtushenko

Critical Propagators in Power Law Banded RMT:Critical Propagators in Power Law Banded RMT:from multifractality to Lfrom multifractality to Léévy flights vy flights

In collaboration with: Philipp Snajberk (LMU & ASC, Munich)Vladimir Kravtsov (ICTP, Trieste)

LMU

Outline of the talkOutline of the talk

1. Introduction: Unconventional RMT with fractal eigenstates Unconventional RMT with fractal eigenstates Multifractality and Scaling properties of correlation Multifractality and Scaling properties of correlation functionsfunctions

2. Strong multifractality regime: Basic ideas of SuSy Virial ExpansionBasic ideas of SuSy Virial Expansion Application of SuSyVE for generalized diffusion propagatorApplication of SuSyVE for generalized diffusion propagator

3. Scaling of retarded propagator: Results and Discussion: Results and Discussion:

multifractality, Lévy flights, phase correlations multifractality, Lévy flights, phase correlations

4. Conclusions and open questionsBielefeld, 16 December 2011

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

n m

localizedlocalized

Model for insulatorsModel for insulators

2 0d

Model for systems at the critical pointModel for systems at the critical point

20 1d

Fractal eigenstates

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 the fractal eigenstates

If ω> then must play a role of L:

L

Lω

ω

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

Two point correlation function:

Critical correlations (Wegner, 1985; Chalker, Daniel, 1988; Chalker 1980)

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

Correlations of the fractal eigenstates

d –space dimension, - mean level spacing, l – mean free path, <…> - disorder averaging

Dynamical scaling hypothesis:Dynamical scaling hypothesis:

Two point correlation function:

Critical correlations (Wegner, 1985; Chalker, Daniel, 1988; Chalker 1980)

(Cuevas , Kravtsov, 2007)

extended

localizedcritical

Fractal enhancement of correlations

Dynamical scaling:

Extended: small amplitudesubstantial overlap in space

Localized: high amplitudesmall overlap in space

the fractal wavefunctions the fractal wavefunctions strongly overlap in spacestrongly overlap in space

Fractal: relatively high amplitude and

- Enhancement of correlations

(The Anderson model: tight binding Hamiltonian)

k

2

n k

nm

Naïve expectation:

weak space correlations

Strong MF regime: 1) do eigenstates really overlap in space?

- sparse fractals

k

2

n k

nm

A consequence of the dynamical scaling:

strong space correlations

Numerical evidence:IQH WF: Chalker, Daniel (1988),

Huckestein, Schweitzer (1994), Prack, Janssen, Freche (1996)Anderson transition in 3d: Brandes, Huckestein, Schweitzer (1996)WF of critical RMTs: Cuevas, Kravtsov (2007)

Strong MF regime: 1) do eigenstates really overlap in space?

First analytical proof for the critical PLBRMT: (V.E. Kravtsov, A. Ossipov, O.Ye., 2010-2011)

- strong MF

– averaged return probability for a wave packet

- spatial scaling (IPR)

1 N

- dynamical scaling

ExpectedExpected scaling scaling properties of properties of P(t)P(t)

P

- IR cutoff of the theory

AnalyticalAnalytical results: results:

Strong MF regime: 2) are MF correlations phase independent?

Retarded propagator of a wave-packet (density-density correlation function):

Diffusion propagator in disordered systems with extended states (“Diffuson”)

Strong MF regime: 2) are MF correlations phase independent?

Retarded propagator of a wave-packet (density-density correlation function):

Dynamical scaling hypothesis for generalized diffuson:

The same scaling exponent The same scaling exponent is possible if phase correlations are noncriticalis possible if phase correlations are noncritical

Q – momentum, - DoS

We calculate for this model

1) to check the hypothesis of dynamical scaling,

2) to study phase correlations in different regimes.

Our current project

Almost diagonal critical PLBRMT from the GOE and GUE symmetry classes

b/|i-j|<<11

- small band width → almost diagonal MF RMT, strong multifractalityalmost diagonal MF RMT, strong multifractality

As an alternative to the σσ-model-model, we use the virial expansion in the number of interacting energy levelsthe virial expansion in the number of interacting energy levels.

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 allows one to expand correlations functions in powers of b<<1 (O.Ye., V.E. Kravtsov, 2003-2005);

Note: a field theoretical machinery of the –model cannot be used in the caseof the strong fractality

SuSy breaking factor

SuSy virial expansion

SuSy is used to average over disorder (O.Ye., Ossipov, Kronmüller, 2007-2009)

m

n

Hmn

Interaction of energy levels

Hybridization of localized stated

m n

Hmn

m n

Hmn

Coupling of supermatrices

Summation over all possible configurationsSummation over all possible configurations

Virial expansion for generalized diffusion propagator

contribution from thediagonal part of RMT

contribution fromj independent supermatrices

Coordinate-time representation:

The probability conservation:

the sum rule:

Note: → critical MF scaling is expected in

Leading term of VE for generalized diffusion propagator

2-matrix approximation:

Extended states

Critical statesat strong multifractality

r d

r 2 d

r2

2 Dt

Levi flightsD iffusion

40 20 20 40

0.2

0.4

0.6

0.8

1.0

Extendedstates

Diffusion

Critical statesat strong multifractality

Lévy flight

Propagators at fixed time(r’=0)

Leading term of VE: MF scaling

Regime of dynamical scaling:

Using the sum rule:

First conclusions• We have found a signature of (strong) multifractality with the scaling exponent = d2 ;• Dynamical scaling hypothesis has been confirmed for the generalized diffuson;• Phases of wave-functions do not have critical correlations.

Leading term of VE: Lévy flights

Beyond regime of MF scaling:

Meaning of - L- Lévi flights évi flights (due to long-range hopping)(due to long-range hopping)

(heavy tailed probability of long steps → power-low tails in the probablity distribution of random walks)

slow decay of correlations because of similar (but not MF) correlations of amplitudes and phases of (almost) localized states

- perturbative in

Beyond leading term of VE: expected log-correcition

Regime of dynamical scaling:

due to dynamical scaling

Subleading terms of the VE are needed to check this guess

Beyond leading term of VE: expected log-correcition

Note: we expect that both exponents are renormalized with increasing b

correlated phases → uncorrelated phases (WD-RMT)

Regime of Lévy flights:

due to decorrelated phases

Subleading term of VE for generalized diffusion propagator

Result 3-matrix approximation (GUE):

Regime of Lévy flights:

No log-corrections fromNo log-corrections from

obeys the sum rule:

Regime of dynamical scaling:

As we expectedAs we expected

No long-rangecorrelations

Phase correlations: crossover “strong-weak” MF regimes

Two different scenarios of the crossover/transition

Non-perturbative (Kosterlizt-Thouless

transition)

LLévy flights exist only at évy flights exist only at b< bb< bcc

Perturbative (smooth crossover)

LLévy flights exist at any finite évy flights exist at any finite bb

Can we expect Lévy flights in short-ranged critical models?

Smooth crossover

Guess: the LGuess: the Lévy flights may exist in both regimes (strong- and évy flights may exist in both regimes (strong- and weak- multifractaity) and all (long-ranged and short-ranged) weak- multifractaity) and all (long-ranged and short-ranged)

critical models. critical models.

Critical RMT vs. Anderson model

Strong multifractality

Weak multifractality

Instability of high gradients operators (Kravtsov, Lerner, Yudson, 1989)

- a precursor of the Lévy flights in AMto be studied

The Lévy flights exist

Conclusions and further plans

• We have demonstrated validity of the dynamical scaling hypothesis for the retarded propagator of the critical almost diagonal RMT.

• Multifractal correlations and Lévy flights co-exist in the case of critical RMT.

• We expect that the Lévy flights exist in regimes of strong- and weak- multifractality in long- and short- ranged critical models.

• We plan to support our hypothesis by performing numerics at intermediate- and -model calculations at large- band width.