Transcript Power-law banded random matrices

On the Multifractal Dimensions
at the Anderson Transition
Imre
Imre Varga
Varga
Departament
Elméleti FizikadeTanszék
Física Teòrica
Universitat
Budapesti Műszaki
de Budapest
és Gazdaságtudományi
de Tecnologia i
Economia,
HongriaBudapest, Magyarország
Egyetem, H-1111
Coauthors: José Antonio Méndez-Bermúdez,
Amando Alcázar-López
(BUAP, Puebla, México)
thanks to : OTKA, AvH
Outline
 Introduction
 The Anderson transition
 Essential features of multifractality
 Random matrix model: PBRM
 Heuristic relations for generalized dimensions
 Spectral compressibility vs. multifractality
 Wigner-Smith delay time
 Further tests
 Conclusions and outlook

Anderson’s model (1958)
 Hamiltonian
 Energies en uncorrelated, random numbers from uniform
(bimodal, Gaussian, Cauchy, etc.) distribution 
W
 Nearest-neighbor „hopping”  V (symmetries: R, C, Q)
 Bloch states for
W V, localized states for W  V
W V
?
Anderson localization
Billy et al. 2008
Hu et al. 2008
Sridhar 2000
Jendrzejewski et al. 2012
Spectral statistics
0<𝜒<1

W < Wc
• extended states
• RMT-like: 𝜒 = 0

• localized states
• Poisson-like: 𝜒 = 1
Σ 2 (𝐿)
𝜒 = lim
𝐿→∞ 𝐿
𝜒 = 0 RMT
W > Wc

W = Wc
• multifractal states
• intermediate stat.
‘mermaid’
semi-Poisson
0<𝜒<1
Eigenstates at small and large W
Extended state
Weak disorder, midband
Localized state
Strong disorder, bandedge
(L=240) R.A.Römer
Multifractal eigenstate at the critical point
http://en.wikipedia.org/wiki/Metal-insulator_transition (L=240) R.A.Römer
Multifractal eigenstate at the critical point
 Inverse participation ratio
• higher precision
• scaling with L
 Box-counting technique
• fixed L
• „state-to-state” fluctuations
 PDF analyzis
Multifractal eigenstate at the critical point
Do these states exist at all?
Yes
Multifractal states in reality
LDOS változása a QH átmeneten keresztül n-InAs(100) felületre elhelyezett Cs réteggel
Multifractal states in
reality
LDOS fluctuations in the vicinity of the
metal-insulator transition Ga1-xMnxAs
Multifractality in general
 Turbulence (Mandelbrot)
 Time series – signal analysis
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Earthquakes
ECG, EEG
Internet data traffic modelling
Share, asset dynamics
Music sequences
etc.
Common features
 Complexity
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Human genome
Strange attractors
etc.

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self-similarity across many scales,
broad PDF
muliplicative processes
rare events
Multifractality in general
 Very few analytically known 𝐷 𝑞

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binary branching process
1d off-diagonal Fibonacci sequence
Baker’s map
etc
 Numerical simulations
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Perturbation series (Giraud 2013)
Renormalization group - NL M – SUSY (Mirlin)
 Heuristic arguments
Numerical multifractal analysis
Parametrization of wave function intensities
2
The set of points where Ψ𝑗 scales with 𝛼
𝜇𝑘 ℓ =
Ψ𝑗
2
Scaling:
• Box size:
ℓ⟶0
• System size: 𝐿 ⟶ ∞
2
∼ 𝐿−𝛼
𝒩𝛼 ∼ 𝐿𝑓(𝛼)
⟹ 𝐼𝑞 ℓ =
𝑗∈box𝑘
ℓ
𝐼𝑞 (ℓ) ∝
𝐿
Ψ𝑗
𝜇𝑘 (ℓ)
𝑘
𝜏(𝑞)
⟹
𝛼𝑞 = 𝜏′(𝑞)
𝑓 𝛼𝑞 = 𝑞𝛼𝑞 − 𝜏(𝑞)
Averaging:
• Typical:
• Ensemble:
exp ln 𝐼𝑞 (ℓ)
𝐼𝑞 (ℓ)
𝑞
Numerical multifractal analysis
Parametrization of wave function intensities
2
The set of points where Ψ𝑗 scales with 𝛼
Ψ𝑗
2
∼ 𝐿−𝛼
𝒩𝛼 ∼ 𝐿𝑓(𝛼)
2
• Ψ𝑗 ≤ 1 ⟹ 𝛼 ≥ 0
• 𝑓 𝛼 convex 𝑓 𝛼0 > 𝑑 = 𝑑
• 𝑓 𝛼1 = 𝛼1
•
Symmetry (Mirlin, et al. 06)
𝛼𝑞 + 𝛼1−𝑞 = 2𝑑
𝑓 2𝑑 − 𝛼 = 𝑓 𝛼 + 𝑑 − 𝛼
Numerical multifractal analysis
Generalized inverse participation number, Rényi-entropies
𝐼𝑞 ∼
𝐿0
𝐿−𝜏(𝑞)
𝐿−𝑑(𝑞−1)
1
𝑅𝑞 =
ln 𝐼𝑞 ∼ 𝐷𝑞 ln 𝐿
1−𝑞
Mass exponent, generalized dimensions
𝜏𝑞 = 𝑑 𝑞 − 1 + Δ𝑞 ≡ 𝐷𝑞 𝑞 − 1
⟹ Δ𝑞 = Δ1−𝑞
Wave function statistics
𝒫(ln 𝜓 2 ) ∼
ln 𝜓 2
𝑓
−𝑑
𝐿 ln 𝐿
parabolic 𝑓 𝛼
log-normal 𝒫(𝑥)
Numerical multifractal analysis
𝜇𝑘 ℓ =
Ψ𝑗
2
𝑗∈box𝑘
ln 𝜇
𝛼=
ln 𝜆
ℓ
if 𝜆 =
𝐿
𝒫(𝛼) ∼ 𝜆𝑑−𝑓(𝛼)
application to quantum percolation, see poster by L. Ujfalusi
Rodriguez et al. 2010
Correlations at the transition
Interplay of eigenvector and spectral correlations
 q=2, Chalker et al. 1995
 q=1, Bogomolny 2011
Cross-correlation of multifractal eigenstates
𝐷𝑞
𝑞𝜒 +
=1
𝑑
Enhanced SC 𝑇𝐶
Auto-correlation of multifractal eigenstates
Feigel’man 2007
Burmistrov 2011
New Kondo phase
Kettemann 2007
Effect of multifractality (PBRM)
Generalize!
Take the model of the model!
PBRM
(a random matrix model)
PBRM:
Power-law Band Random Matrix
 model:
matrix,
 asymptotically:
 free parameters
and
b
Mirlin, et al. ‘96, Mirlin ‘00
PBRM

Mirlin, et al. ‘96, Mirlin ‘00
 𝑏 ≫ 1 weak
multifractality
𝐷2 ⟶ 1 χ ⟶ 0
 𝑏 ≪ 1 strong
multifractality
𝐷2 ⟶ 0 χ ⟶ 1
𝜒~ 1 + 𝑎𝜒 𝑏
−1
Mirlin, et al. ‘96, Mirlin ‘00
Generalized dimensions
𝐷𝑞 ≈ 1 + 𝑎𝑞 𝑏
𝐷𝑞 ′
−1 −1
𝑞𝐷𝑞
≈ ′
𝑞 + 𝑞 − 𝑞′ 𝐷𝑞
JAMB és IV (2012)
PBRM at criticality (0.5 < 𝑞 < 5)
General relations
𝑞𝐷𝑞
≈ const ~ 𝑏
1 − 𝐷𝑞
Spectral statistics and 𝐷𝑞
1 − 𝐷𝑞
𝜒≈
1 + (𝑞 − 1)𝐷𝑞
e.g.:
𝐷1
1−𝜒
𝐷2 ≈
≈
2 − 𝐷1 1 + 𝜒
JAMB és IV (2012)
Higher dimensions, 𝑞 < 0.5
Replace
𝐷𝑞
𝐷𝑞 →
𝑑
2dQHT
For 𝑞 < 0.5 using Δ𝑞 = Δ1−𝑞
3dAMIT
𝐷𝑞 ≈
1 − 2𝑞
𝑞
𝐷1
+
1−𝑞
1 − 𝑞 1 + 𝑞(𝐷1 − 1)
1 − 𝐷𝑞
𝜒≈
𝑞(2 − 𝐷𝑞 )
JAMB és IV (2012)
Surpisingly robust and general
Different problems
𝑞𝐷𝑞
≈ const
1 − 𝐷𝑞
1 − 𝐷𝑞
𝜒≈
1 + (𝑞 − 1)𝐷𝑞
 Random matrix ensembles
 Ruijsenaars-Schneider ensemble
 Critical ultrametric ensemble
 Intermediate quantum maps
 Calogero-Moser ensemble
 Chaos
 baker’s map
 Exact, deterministic problems
 Binary branching sequence
 Off-diagonal Fibonacci sequence
JAMB és IV (2013)
Scattering: system + lead
 Scattering matrix
 Wigner-Smith delay time
 Resonance widths:
eigenvalues of
poles of
Scattering: PBRM + 1 lead
 JA Méndez-Bermúdez – Kottos ‘05
Ossipov – Fyodorov ‘05:
 JA Méndez-Bermúdez – IV 06:
Scattering exponents
Wigner-Smith delay time
exp ln 𝜏 ∼ 𝐿𝜎𝜏
𝜎𝜏 = 1 − 𝜒
𝜏 −𝑞 ~𝐿−𝜎𝑞
𝜎𝑞 =
𝑞(1 − 𝜒)
1 + 𝑞𝜒
JAMB és IV (2013)
Summary and outlook
 Multifractal states in general
 Random matrix model (PBRM)
 heuristic relations tested for many models, quantities
 New physics involved
 Kondo, SC, graphene, etc.
 Outlook
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Interacting particles (cf. Mirlin et al. 2013)
Decoherence
Proximity effect (SC)
Topological insulators
Thanks for your attention