Imre Varga Elméleti Fizika TanszékBudapesti Műszaki és Gazdaságtudományi Egyetem, H-1111 Budapest, Magyarország
Imre VargaDepartament de Física TeòricaUniversitat de Budapest de Tecnologia i Economia, Hongria
On the Multifractal Dimensions at the Anderson Transition
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
RMT
0<𝜒<1
𝜒= lim𝐿→ ∞
Σ 2(𝐿)𝐿
W < Wc• extended states• RMT-like:
W > Wc
• localized states• Poisson-like:
W = Wc• multifractal states• intermediate stat.
‘mermaid’ semi-Poisson
Eigenstates at small and large W
Extended stateWeak disorder, midband
Localized stateStrong 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
Box-counting technique• fixed L• „state-to-state” fluctuations
PDF analyzis
• higher precision• scaling with L
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
Earthquakes ECG, EEG Internet data traffic modelling Share, asset dynamics Music sequences etc.
Complexity Human genome Strange attractors etc.
Common features self-similarity across many scales, broad PDF muliplicative processes rare events
Multifractality in general
Very few analytically known binary branching process 1d off-diagonal Fibonacci sequence Baker’s map etc
Numerical simulations Perturbation series (Giraud 2013) Renormalization group - NLM – SUSY (Mirlin)
Heuristic arguments
Numerical multifractal analysis
Parametrization of wave function intensities |Ψ 𝑗|2∼𝐿−𝛼
The set of points where scales with 𝒩𝛼∼ 𝐿𝑓 (𝛼 )
Scaling:• Box size: • System size:
Averaging:• Typical: • Ensemble:
𝜇𝑘 ( ℓ )= ∑𝑗∈ box𝑘
|Ψ 𝑗|2⟹𝐼𝑞 (ℓ )=∑
𝑘[𝜇𝑘(ℓ) ]𝑞
𝐼𝑞(ℓ)∝( ℓ𝐿 )
𝜏 (𝑞)
⟹ { 𝛼𝑞=𝜏 ′(𝑞)𝑓 (𝛼𝑞)=𝑞𝛼𝑞−𝜏 (𝑞)
Numerical multifractal analysis
Parametrization of wave function intensities |Ψ 𝑗|2∼𝐿−𝛼
The set of points where scales with 𝒩𝛼∼ 𝐿𝑓 (𝛼 )
• convex
• Symmetry (Mirlin, et al. 06)
Numerical multifractal analysis
Generalized inverse participation number, Rényi-entropies
Mass exponent, generalized dimensions
Wave function statistics
⟨ 𝐼𝑞 ⟩∼ { 𝐿0
𝐿−𝜏 (𝑞)
𝐿−𝑑(𝑞− 1)𝑅𝑞=
11−𝑞
ln ⟨ 𝐼𝑞 ⟩∼𝐷𝑞 ln 𝐿
𝜏𝑞=𝑑 (𝑞−1 )+Δ𝑞≡𝐷𝑞 (𝑞−1 )⟹ Δ𝑞=Δ1 −𝑞
𝒫(ln|𝜓|2)∼ 𝐿𝑓 ( ln|𝜓|2
ln 𝐿 )−𝑑parabolic og-normal
Numerical multifractal analysis
Rodriguez et al. 2010
𝜇𝑘 ( ℓ )= ∑𝑗∈ box𝑘
|Ψ 𝑗|2
𝛼=ln𝜇ln 𝜆
if 𝜆=ℓ𝐿
𝒫(𝛼)∼ 𝜆𝑑− 𝑓 (𝛼)
application to quantum percolation, see poster by L. Ujfalusi
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
Auto-correlation of multifractal eigenstates
𝑞 𝜒+𝐷𝑞
𝑑=1
Enhanced SC Feigel’man 2007Burmistrov 2011
New Kondo phaseKettemann 2007
Effect of multifractality (PBRM)
Generalize!
Take the model of the model!
PBRM (a random matrix model)
b
PBRM: Power-law Band Random Matrix
model: matrix,
asymptotically:
free parameters and
Mirlin, et al. ‘96, Mirlin ‘00
PBRM
Mirlin, et al. ‘96, Mirlin ‘00
Mirlin, et al. ‘96, Mirlin ‘00
weak multifractality
strong multifractality
𝜒 (1+𝑎𝜒 𝑏)− 1
JAMB és IV (2012)
Generalized dimensions
𝐷𝑞≈ [ 1+(𝑎𝑞𝑏)−1 ]− 1
𝐷𝑞 ′ ≈𝑞𝐷𝑞
𝑞′+(𝑞−𝑞′ )𝐷𝑞
JAMB és IV (2012)
General relations
𝑞𝐷𝑞
1−𝐷𝑞
≈ const 𝑏
Spectral statistics and
𝜒 ≈1−𝐷𝑞
1+(𝑞−1)𝐷𝑞
𝐷2≈𝐷1
2−𝐷1
≈1− 𝜒1+𝜒
PBRM at criticality
e.g.:
JAMB és IV (2012)
Replace 𝐷𝑞→𝐷𝑞
𝑑
For using
𝜒 ≈1−𝐷𝑞
𝑞 (2−𝐷𝑞)
Higher dimensions,
2dQHT
3dAMIT
𝐷𝑞≈1− 2𝑞1−𝑞
+ 𝑞1−𝑞 ( 𝐷1
1+𝑞 (𝐷1 −1))
JAMB és IV (2013)
Different problems
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
𝑞𝐷𝑞
1−𝐷𝑞
≈ const 𝜒 ≈1−𝐷𝑞
1+(𝑞−1)𝐷𝑞
Surpisingly robust and general
Scattering: system + lead Scattering matrix
Wigner-Smith delay time
Resonance widths: eigenvalues of poles of
JA Méndez-Bermúdez – Kottos ‘05 Ossipov – Fyodorov ‘05:
JA Méndez-Bermúdez – IV 06:
Scattering: PBRM + 1 lead
JAMB és IV (2013)
Wigner-Smith delay time
Scattering exponents
⟨𝜏−𝑞 ⟩ 𝐿−𝜎 𝑞exp ⟨ ln𝜏 ⟩∼ 𝐿𝜎 𝜏
𝜎 𝜏=1− 𝜒 𝜎 𝑞=𝑞 (1− 𝜒 )1+𝑞 𝜒
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 Interacting particles (cf. Mirlin et al. 2013) Decoherence Proximity effect (SC) Topological insulators
Thanks for your attention