oleg yevtushenko critical scaling in random matrices with fractal eigenstates in collaboration with:...
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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
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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
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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)
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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
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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)
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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)
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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
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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
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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
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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
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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
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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
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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)
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Part I: Calculations
Part II: Scenarios and speculations
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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
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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)
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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
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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
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Part I: Calculations
Part II: Scenarios and speculations
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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.
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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:
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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)
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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
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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
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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”
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* *
/
*
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