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