Post on 31-Jan-2016
description
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.