Transcript Document

Ludwig-Maximilians-Universität, München, Germany
Arnold Sommerfeld Center for Theoretical Physics
Critical scaling in Random Matrices
with fractal eigenstates
Oleg Yevtushenko
In collaboration with:
Vladimir Kravtsov (ICTP, Trieste),
Alexander Ossipov (University of Nottingham)
Emilio Cuevas (University of Murcia)
Brunel, 18 December 2009
Outline of the talk
1. Introduction:


Unconventional RMT with fractal eigenstates
Local Spectral correlation function: Scaling properties
2. Strong multifractality regime:


Virial Expansion: basic ideas
Application of VE for critical exponents
3. Scaling exponents:


Calculations: Contributions of 2 and 3 overlapping eigenstates
Speculations: Scenario for universality and Duality
4. Conclusions
Brunel, 18 December 2009
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
0  d2  d
(fractal) dimension of a support:
the space dimension d=1 for RMT
0  d2  1
d2  1
 n k 
d2
 n k 
2
extended (WD)
n
n
d2  0
 n k 
2
fractal
m
m
k
Model for metals
k
Model for systems
at the critical point
2
n
localized
m
k
Model for insulators
MF RMT: Power-Law-Banded Random Matrices
Hi, j
2


b is the bandwidth
1
1 i  j
b

 1, i  j  b

2
| H i , j |2 ~ 
 1i  j , i  j  b


2

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 MF eigenstates
Local in space two point spectral correlation function (LDOS-LDOS):
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)
d –space dimension, L – system size,  - mean level spacing
MF enhancement of eigenstate correlations: the Anderson model
The Anderson model: tight binding Hamiltonian of a disordered system (3-diagonal RM)
The Chalkers’ scaling:
- Enhancement of correlations
Extended states:
small amplitude
high probability of overlap in space
localized
Localized states:
high amplitude
small probability of overlap in space
critical
extended
MF states:
relatively high amplitude and
the fractal eigenstates
strongly overlap in space
(Cuevas , Kravtsov, 2007)
Universality of critical correlations: MF RMT vs. the Anderson model
Anderson model at criticality (MF eigenstates),
dimension d
MF (critical) RMT,
bandwidth b
“” – MF PLBRMT, β=1, b = 0.42
“” – 3d Anderson model from orthogonal class
with MF eigenstates at the mobility edge, E=3.5
(Cuevas, Kravtsov, 2007)
Advantages of the critical RMT:
1) numerics are not very time-consuming;
2) it is known how to apply the SuSy field theory
Strong MF regime: do eigenstates really overlap in space?
- sparse fractals
A consequence of the Chalker’s scaling:
Naïve expectation:
 n k 
n
 n k 
2
m
n
2
m
k
weak space correlations
k
strong space correlations
So far, no analytical check of the Chalker’s scaling; just a numerical evidence
Statement of problem
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
H ii
2
1
 ,
2
Hi j
2
1

2

2
1
1 i  j
b

2
Almost diagonal RMT from the GUE symmetry class
1 b 
 
 , b  1
2i j 
Method: The virial expansion
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 correlation functions in powers of b<<1
VE for RMT: 1) the Trotter formula & combinatorial analysis (OY, Kravtsov, 2003-2005);
2) Supersymmetric FT (OY, Ossipov, Kronmueller, 2007-2009).
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
Expected scaling properties
- the IPR spatial scaling
- the Chalker’s dynamical scaling
( - scaled time)
VE for the return probability:
VE for the scaling exponent
O(b1)
O(b2)
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
Discrete system: summation over 1d lattice in all terms of the VE
small distances are regularized by the lattice
Assumption: the scaling exponents are not sensitive to small distances
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 leading term of the virial expansion
Homogeneity of the argument at  → 0
The scaling assumption and the relation =1-d2 hold true up to O(b)
Three level contribution
The subleading term of the virial expansion
homogeneous arguments β/x and β/y at →0
Calculations:
cancel out in P(3)- (P(2))2/2
The scaling assumption holds true up to the terms of order O(b2 log2())
Part I: Calculations
Part II: Scenarios and speculations
Is the Chalker’s relation =1-d2 exact?
Regime of strong multifractality
Intermediate regime
(Cuevas, O.Ye., unpublished)
(Cuevas, Kravtsov, 2007)
Numerics confirm that the Chalker’s
relation is exact and holds true for any b.
Which conditions (apart from the homogeneity property) are
necessary to prove universality of subleading terms of order
O( b2 ) in the scaling exponents?
Universality of the scaling exponent
Sub-leading
contributions to the scaling exponents:
Integral representations:
The homogeneity property results in:
Assumption: the scaling exponents do not contain anomalous
contributions (coming from uncertainties
then
The Chalker’s relation =1-d2 holds true up to O(b2)
(a hint that it is exact)
)
Duality of scaling exponents: small vs. large b-parameter
- Strong multifractality (b << 1)
- Weak multifractality (b >> 1)
Note that
at B<<1 & at B >>1
Does this equality hold true only at small-/large- or at 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
strong and weak multifractality!?
(Kravtsov, arXiv:0911.06, Kravtsov, Cuevas, O.Ye., Ossipov [in progress] )
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 true
up to the terms of order O(b2 log2())
• We have proven that the Chalker’s relation =1-d2 holds true up to the terms
of order O(b)
• We have suggested a schenario which (under certain assumptions) expains
why the Chalker’s relation =1-d2 holds true up to the terms of order O(b2) –
a hint that the Chalker’s relation is exact
• We plan
a) to generalize the results accounting for an interaction
of arbitrary number of levels
b) to study duality in the RMT with multifractal eigenstates
The supersymmetric action for RMT

R
/ A
ˆ


G (E ) 
 
commuting
variables
1
Supermatrix:
- Retarded/Advanced
Green’s functions
Q j (resolvents):
 j  j
variables
E anticommuting
 Hˆ  i 0
breaks SuSy in R/A sectors
One-matrix part of action
  diag (1, 1) RA
a
Weak “interaction” of supermatrices
1
H aa2
2
SuSy virial expansion for almost diagonal RMs
Let’s rearrange
“interacting part”
V
D
Diagonal part of RMT
Qm
Localized eigenstates →
etc.
noninteracting
Q-matrices
Q
(Mayer’s
function)
n
…
m
n
Perturbation theory
in off-diagonal matrix elements
V
(2)
V
(3,4, )
Qm
m
H mn
…
2
Qn
Interaction of 2 matrices
(of 2 localized eigenstates)
n
Subleading terms: Interaction of 3, 4 … matrices
Method: Green’s functions and SuSy representation
Gˆ R / A ( E ) 
1
E  Hˆ  i 0
- Retarded/Advanced Green’s functions (resolvents):
How to average over disorder?
The problem of denominator:
1
1
*trick* Z 
The supersymmetry
d d   exp iS
  dd * expiS (  )
  d d exp

iS ( )

R/ A
S   ij i ( E ij  H ij ) j
Gij

*
d

d

exp iS 

R/ A
Gij ( E ,  )   d  d  i j * exp iS  ( )  iS  (  ) 
i
j
*

S ( )   ij i ( E ij  H ij ) j ; S (  )   ij i ( E ij  H ij )  j