Critical Propagators in Power Law Banded RMT:
from multifractality to Lévy flights
Oleg Yevtushenko
In collaboration with:
Philipp Snajberk (LMU & ASC, Munich)
Vladimir Kravtsov (ICTP, Trieste)
LMU
Outline of the talk
1. Introduction:


Unconventional RMT with fractal eigenstates
Multifractality and Scaling properties of correlation functions
2. Strong multifractality regime:


Basic ideas of SuSy Virial Expansion
Application of SuSyVE for generalized diffusion propagator
3. Scaling of retarded propagator:

Results and Discussion:
multifractality, Lévy flights, phase correlations
4. Conclusions and open questions
Bielefeld, 16 December 2011
WD and Unconventional Gaussian RMT
The Schrödinger equation for a 1d chain:
(eigenvalue/eigenvector problem)
Hˆ n  n n , Hˆ  Hˆ †
H - Hermithian matrix with random (independent, Gaussian-distributed) entries
Statistics of RM entries:
 H ij  0,  H 
2
ii
1

2
,  H i  j  F (i  j )
If F(i-j)=1/2 → the Wigner–Dyson (conventional) RMT (1958,1962):
Parameter β reflects symmetry classes (β=1: GOE, β=2: GUE)
F(x)
Generic unconventional RMT:
A
1
x
Function F(i-j) can yield universality classes of the eigenstates, different from WD RMT
Factor A parameterizes spectral statistics and statistics of eigenstates
3 cases which are important for physical applications
4
Inverse Participation Ratio P2  
1
  k    E  
n
n
n,k
P2
N 
 1
N
d2  1
 n k 
d2
0  d2  d
(fractal) dimension of a support:
the space dimension d=1 for RMT
0  d2  1
2
Fractal eigenstates
extended (WD)
n
d2  0
 n k 
2
n
m
k
localized
m
k
Model for systems at the critical point
Model for insulators
Model for metals
MF RMT: Power-Law-Banded Random Matrices
Hi, j
2


b is the bandwidth
1
1 i  j
b

| H i , j |2
2
 1, i  j  b

2
~
 1i  j , i  j  b



RMT with multifractal eignestates at any band-width
(Mirlin, Fyodorov et.al., 1996, Mirlin, Evers, 2000 - GOE and GUE symmetry classes)
2 π b>>1
d 2  1  const
b<<1
2 b
1-d2<<1 – regime of weak multifractality
d2  const b
d2<<1 – regime of strong multifractality
Correlations of the 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)
If ω> then
must play a role of L:
ω

Lω
L
Correlations of the 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)
Dynamical scaling hypothesis:
d –space dimension,  - mean level spacing, l – mean free path, <…> - disorder averaging
Fractal enhancement of correlations
Dynamical scaling:
- Enhancement of correlations
Extended:
localized
small amplitude
substantial overlap in space
critical
extended
Localized:
high amplitude
small overlap in space
Fractal:
relatively high amplitude and
(Cuevas , Kravtsov, 2007)
(The Anderson model: tight binding Hamiltonian)
the fractal wavefunctions
strongly overlap in space
Strong MF regime: 1) do eigenstates really overlap in space?
- sparse fractals
Naïve expectation:
 n k 
n
A consequence of the dynamical scaling:
 n k 
2
m
n
2
m
k
weak space correlations
k
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
Expected scaling
properties of P(t)
P
- dynamical
scaling
1
Analytical results:
- IR cutoff of the theory
- spatial
scaling (IPR)
N
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:
Q – momentum,  - DoS
The same scaling exponent is possible if phase correlations are noncritical
Our current project
Almost diagonal critical PLBRMT from the GOE and GUE symmetry classes
1
b/|i-j|<<1
- small band width → almost diagonal MF RMT, strong multifractality
We calculate
for this model
1) to check the hypothesis of dynamical scaling,
2) to study phase correlations in different regimes.
Method: The virial expansion
As
an alternative
to the σ-model,
use–model cannot be used in the case
Note:
a field theoretical
machinery we
of the
of the expansion
strong fractality
the virial
in the number of interacting energy levels.
Gas of low density ρ
Almost diagonal RM
  b
bΔ
ρ1
b1
Δ
2-particle collision
2-level interaction
b2
ρ2
3-particle collision
3-level interaction
VE allows one to expand correlations functions in powers of b<<1
(O.Ye., V.E. Kravtsov, 2003-2005);
SuSy virial expansion
Hybridization of localized stated
Interaction of energy levels
m
Hmn
n
Hmn

m
n
SuSy is used to average over disorder (O.Ye., Ossipov, Kronmüller, 2007-2009)
Hmn
m

Coupling of supermatrices
n
Summation over all possible configurations
SuSy breaking factor
Virial expansion for generalized diffusion propagator
Coordinate-time representation:
contribution from the
diagonal part of RMT
contribution from
j independent supermatrices
The probability conservation:
the sum rule:
Note:
→ critical MF scaling is expected in
Leading term of VE for generalized diffusion propagator
2-matrix approximation:
1.0
Propagators at fixed time
(r’=0)
0.8
CriticalCritical
states
states
at
m ultifractality
atstrong
strong
multifractality
0.6
Extended
Extended
states
states
0.4
Diffusion
Diffusion
r
Lévy
flight
Levi
flights
d
0.2
r2
2 Dt
r
40
20
2d
20
40
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:
- perturbative in
slow decay of correlations because of similar
(but not MF) correlations of amplitudes
and phases of (almost) localized states
Meaning of
- Lévi flights (due to long-range hopping)
(heavy tailed probability of long steps →
power-low tails in the probablity distribution of random walks)
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
Regime of Lévy flights:
correlated phases
→
uncorrelated phases
(WD-RMT)
due to decorrelated phases
Note: we expect that both exponents are renormalized with increasing b
Subleading term of VE for generalized diffusion propagator
Result 3-matrix approximation (GUE):
obeys the sum rule:
Regime of dynamical scaling:
Regime of Lévy flights:
No log-corrections from
As we expected
Phase correlations: crossover “strong-weak” MF regimes
Two different scenarios of the crossover/transition
Non-perturbative
(Kosterlizt-Thouless transition)
No long-range
correlations
Lévy flights exist only at b< bc
Perturbative
(smooth crossover)
Lévy flights exist at any finite b
Can we expect Lévy flights in short-ranged critical models?
Smooth crossover
Critical RMT vs. Anderson model
Strong multifractality
The Lévy flights exist
Weak multifractality
to be studied
Instability of high gradients operators
(Kravtsov, Lerner, Yudson, 1989)
- a precursor of the Lévy flights in AM
Guess: the Lévy flights may exist in both regimes (strong- and weakmultifractaity) and all (long-ranged and short-ranged) critical models.
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.