Anderson transitions and wave function multifractality

Part II

Alexander D. Mirlin

Research Center Karslruhe & University Karlsruhe & PNPI St. Petersburg

http://www-tkm.physik.uni-karlsruhe.de/∼mirlin/

Plan (tentative)

• Quantum interference, localization, and field theories of disordered systems

• diagrammatics; weak localization; mesoscopic fluctuations

• field theory: non-linear σ-model

• quasi-1D geometry: exact solution, localization

• RG, metal-insulator transition, criticality

•Wave function multifractality

• wave function statistics in disordered systems

• multifractality of critical wave functions

• properties of multifractality spectra: average vs typical, possible singu-larities, relations between exponents, surface/corner multifractality

• Systems and models

• Anderson transition in D dimensions

• Power-Law Random Banded Matrix model: 1D system with 1/r hopping

• mechanisms of criticality in 2D systems

• quantum Hall transitions in normal and superconducting systems

Evers, ADM, arXiv:0707.4378 “Anderson transitions”, Rev.Mod.Phys., in print

Wave function statistics

Fyodorov, ADM 92; . . . Review: ADM, Phys. Rep. 2000

distribution function P(|ψ2(r)|), correlation functions 〈|ψ2(r1)ψ2(r2)|〉 etc.

perturbative (diagrammatic) approach not sufficient

Field-theoretical method: σ-model Wegner 79, Efetov 82 (SUSY)

S[Q] ∝∫

ddr Str[−D(∇Q(r))2 − 2iωΛQ(r)] Q2(r) = 1

Q ∈ {sphere × hyperboloid} “dressed” by Grassmannian variables

σ-model contains all the diffuson-cooperon diagrammatics + much more (stronglocalization; Anderson transition & RG; non-perturbative effects)

• zero mode (Q = const) −→ RMT distribution P(|ψ2(r)|)ψ(r) – uncorrelated Gaussian random variables

• diffusive modes [ΠD(r1, r2)] −→ deviations from RMT,long-range spatial correlations

parameter g =D/L2

∆≡ Thouless energy

level spacing=

G

e2/hdimensionless conductance

g À 1: metal g ¿ 1: strong localization

quasi-1D: g = ξ/L, ξ – localization length

Experiment:

Wave function statistics in disordered microwave billiards

Kudrolli, Kidambi, Sridhar, PRL 95

Wave function statistics

Distribution P(t) of t = V |ψ2(r)|

Metallic samples (dimensionless conductance g À 1):

• Main body of the distribution:

P(t) = e−t[

1 +κ

2(2− 4t+ t2) + . . .

]

(U)

P(t) =e−t/2√

2πt

[

1 +κ

2

(

3

2− 3t+

t2

2

)

+ . . .

]

(O)

κ = ΠD(r, r) ∝ 1/g ¿ 1 (classical return probability)

• Asymptotic tail:

P(t) ∝

exp{− . . .√t} , quasi-1D

exp{− . . . lnd t} , d = 2, 3

Wave function statistics: numerical verification

numerics: Uski, Mehlig, Romer, Schreiber 01

quasi-1d, metallic regime, distribution of t = V |ψ2(r)|

“body”: 1/g corrections to RMT “tail” ∝ exp(− . . .√t)

Physics of the slowly decaying “tail”: anomalously localized states

Anomalously localized states: imaging

numerics: Uski, Mehlig, Schreiber ’02

spatial structure

of an anomalously localized state (ALS):

〈|ψ(r)|2δ(V|ψ(0)|2 − t)〉

with t atypically large

ADM ’97

ALS determine asymptotic behavior of distributions of various quantities

(wave function amplitude, local and global density of states, relaxation time,. . . )

Altshuler, Kravtsov, Lerner, Fyodorov, ADM, Muzykantskii, Khmelnitskii,Falko, Efetov. . .

Wave function statistics: numerical verification. 2D.

numerics: Uski, Mehlig, Romer, Schreiber ’01

“body” of the distribution:

(1/g) ln(L/l) corrections

“tail” ∝ exp(− . . . ln2 t)

Falko, Efetov ’95

−→ precursors of Anderson criticality

Multifractality at the Anderson transition

Pq =∫

ddr|ψ(r)|2q inverse participation ratio

〈Pq〉 ∼

L0 insulatorL−τq criticalL−d(q−1) metal

τq = d(q − 1) + ∆q ≡ Dq(q − 1) multifractality

normal anomalous ∆0 = ∆1 = 0

Wave function correlations at criticality:

L2d〈|ψ2(r)ψ2(r′)|〉 ∼ (|r− r′|/L)−η , η = −∆2

Ld(q1+q2)〈|ψ2q1(r1)ψ2q2(r2)|〉 ∼ L−∆q1−∆q2(|r1 − r2|/L)∆q1+q2−∆q1−∆q2

many-points correlators 〈|ψ2q1(r1)ψ2q2(r2) . . . ψ

2qn(rn)|〉 — analogously

different eigenfunctions:L2d〈|ψ2

i (r)ψ2j(r′)|〉

L2d〈ψi(r)ψ∗j (r)ψ∗i (r′)ψj(r′)〉

}

∼(|r− r′|

Lω

)−η

ω = εi − εj Lω ∼ (ρω)−1/d ρ – DoS |r− r′| < Lω

Multifractality and the field theory

∆q – scaling dimensions of operators O(q) ∼ (QΛ)q

d = 2 + ε: ∆q = −q(q − 1)ε+ O(ε4) Wegner ’80

• Infinitely many operators with negative scaling dimensions

• ∆1 = 0 ←→ 〈Q〉 = Λ naive order parameter uncritical

Transition described by an order parameter function F (Q)

Zirnbauer 86, Efetov 87

←→ distribution of local Green functions and wave function amplitudes

ADM, Fyodorov ’91

Singularity spectrum

d α0α

d

0

f(α)

metalliccritical

α− α+

|ψ|2 large |ψ|

2 small

τq −→ Legendre transformation

τq = qα− f(α) , q = f ′(α) , α = τ ′q

−→ singularity spectrum f(α)

P(|ψ2|) ∼ 1

|ψ2|L−d+f(− ln |ψ2|

lnL ) wave function statistics

To verify: calculate moments 〈Pq〉 and use saddle-point method

〈Pq〉 = Ld〈|ψ2q|〉 ∼∫

dαL−qα+f(α) , α = − ln |ψ2|/ lnL

Lf(α) – measure of the set of points where |ψ|2 ∼ L−α

• τq – non-decreasing, convex (τ ′q ≥ 0, τ ′′q ≤ 0 ), with τ0 = −d, τ1 = 0

• f(α) – convex (f ′′(α) ≤ 0), defined on α ≥ 0, maximum f(α0) = d.

Statistical ensemble −→ f(α) may become negative

Multifractal wave functions at the Quantum Hall transition

Role of ensemble averaging: Average vs typical spectra

P typq = exp〈lnPq〉 ∼ L−τ

typq ,

τ typq =

qα− , q < q−τq , q− < q < q+

qα+ , q > q+

Singularity spectrum f typ(α) is defined on [α+, α−], where it is equal to f(α).

IPR distribution:

P(Pq/Ptypq ) is scale-invariant at criticality

Power-law tail at large Pq/Ptypq :

P(Pq/Ptypq ) ∝ (Pq/P

typq )−1−xq

tail exponent xq

= 1 , q = q±> 1 , q− < q < q+

< 1 , otherwise

xqτtypq = τqxq

−7 −6 −5 −4 −3ln P2

10−2

10−1

100

Dis

trib

utio

n (ln

P2)

256512102420484096

Dimensionality dependence of multifractality

RG in 2 + ε dimensions, 4 loops, orthogonal and unitary symmetry classes

Wegner ’87

∆(O)q = q(1− q)ε+

ζ(3)

4q(q − 1)(q2 − q + 1)ε4 + O(ε5)

∆(U)q = q(1− q)(ε/2)1/2 − 3

8q2(q − 1)2ζ(3)ε2 + O(ε5/2)

ε¿ 1 −→ weak multifractality

−→ keep leading (one-loop) term −→ parabolic approximation

τq ' d(q − 1)− γq(q − 1), ∆q ' γq(1− q) , γ ¿ 1

f(α) ' d− (α− α0)2

4(α0 − d); α0 = d+ γ

γ = ε (orthogonal); γ = (ε/2)1/2 (unitary)

q± = ±(d/γ)1/2

Dimensionality dependence of multifractality: IPR distribution

Mildenberger, Evers, ADM ’02

−8 −6 −4 −2ln P2

0

0.2

0.4

0.6

0.8

Dis

trib

utio

n(ln

P2)

−5 −3 −1ln P2

a) b)

0 1 2 3q

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

σ q(∞

)

10 100L

0.1

1

5

σ q(L)

65432.521.50.5

IPR distribution in 3D and 4D

3D: L = 8, 11, 16, 22, 32, 44, 64, 80

4D: L = 8, 10, 12, 14, 16

variance σq of distribution P(lnPq)

2 + ε with ε = 0.2, 1 (analytics),

3D, 4D (numerics)

one-loop results in d = 2 + ε:

σq = 8π2aε2q2(q − 1)2 , |q| ¿ q+, a ' 0.00387 (periodic b.c.)

σq ' x−1q = q/q+ , q > q+

Dimensionality dependence of multifractality spectra

0 1 2 3q

0

1

2

3

4D

q

~

0 1 2 3 4 5 6 7α

0

1

2

3

4

−1

f(α)

0 1 2 3 4 5α

0

1

2

3

−1

f(

α)

~

~

Analytics (2 + ε, one-loop) and numerics

τq = (q − 1)d− q(q − 1)ε+ O(ε4)

f(α) = d− (d+ ε− α)2/4ε+ O(ε4)

d = 4 (full)d = 3 (dashed)d = 2 + ε, ε = 0.2 (dotted)d = 2 + ε, ε = 0.01 (dot-dashed)

Inset: d = 3 (dashed)vs. d = 2 + ε, ε = 1 (full)

Mildenberger, Evers, ADM ’02

Singularities in multifractal spectra: Termination and freezing

0 1 2 3α-4

-3

-2

-1

0

1

2

f(α)

(a)

0 1 2 3q

-2

-1

0

1

2

τ q

0 1 2 3α

(b)

0 1 2 3q

0 1 2 3α

(c)

0 1 2 3q

(a) no singularities (b) termination (c) freezing

Relations between multifractal exponents

Non-linear σ-model

−→ distribution of local Green function GR(r, r)/π〈ρ〉 = u− iρ

P (u, ρ) =1

2πρP0

(

(u2 + ρ2 + 1)/2ρ)

−→ symmetry of LDOS distribution: Pρ(ρ) = ρ−3Pρ(ρ−1)

ADM, Fyodorov ’94 (β = 2)

Recently:

more complete derivation (β = 1, 2, 4) via relation to a scattering problem:

system with a channel attached at a point r

S-matrix S = (1− iK)/(1 + iK) K = 12V †GR(r, r)V = u− iρ

distribution P (S) invariant with respect to the phase θ of S =√reiθ.

Fyodorov, Savin, Sommers ’04-05

Relations between multifractal exponents (cont’d)

ADM, Fyodorov, Mildenberger, Evers ’06

LDOS distribution in σ-model + universality

−→ exact symmetry of the multifractal spectrum:

∆q = ∆1−q f(2d− α) = f(α) + d− α

Consequence: assuming no singularities in f(α) spectrum,

its support is bounded by the interval [0, 2d]

−3 −2 −1 0 1 2 3 4q

−3

−2

−1

0

∆ q, ∆

1−q

b=4

b=1

b=0.3b=0.1

0 0.5 1 1.5 2α

−1.5

−1

−0.5

0

0.5

1

f(α)

b=4b=

1

b=0.

3

b=0.1

Relations between multifractal exponents (cont’d):

Anderson transition in symplectic class in 2D

Mildenberger, Evers ’07

Symmetry of the spectrum: ∆q = ∆1−q

Non-parabolicity of the spectrum

δ(q) ≡ ∆q

q(1− q)6= const

Conformal invariance

2D ↔ quasi-1D strip

Λc = 1/πδ0

δ0 ' 0.172 , Λc ' 1.844

−→ πδ0Λc = 0.999± 0.003

Related work:

Obuse, Subramaniam, Furusaki, Gruzberg, Ludwig ’07

Relations between multifractal exponents (cont’d)

Wigner delay time: tW = ∂θ(E)/∂E

Relation between wave function (Py) and delay time (PW ) distribution

in the σ-model:

PW (tW ) = t−3W Py(t−1

W ) tW = tW∆/2π

Ossipov, Fyodorov ’05

+ universality

−→ exact relation between multifractal exponents

for closed (wave functions, τq) and open (delay times, γq) systems

γq = τ1+q

ADM, Fyodorov, Mildenberger, Evers ’06

Surface multifractality

Subramaniam, Gruzberg, Ludwig, Evers, Mildenberger, ADM ’06

Critical fluctuations of wave functions at surface: new set of exponents

Ld−1〈|ψ(r)|2q〉 ∼ L−τ sq τ s

q = d(q − 1) + qµ+ 1 + ∆sq

Weak multifractality (2 + ε or 2D): γ = (βπg)−1¿ 1

τ bq = 2(q − 1) + γq(1− q)

τ sq = 2(q − 1) + 1 + 2γq(1− q)

fb(α) = 2− (α− 2− γ)2/4γ

f s(α) = 1− (α− 2− 2γ)2/8γ

Studied numerically for a variety of crit-ical systems: PRBM, IQHE, SQHE, 2Dsymplectic

1.5 2 2.5α−1

0

1

2

f(α)

α+

s α+

b α−

b α−

s

−20 −10 0 10 20q−4

−2

0

2

τ q−2(

q−1)

q−bs q+

bs

bulk

surface

bulk

surface

Corner multifractality

Obuse, Subramaniam, Furusaki, Gruzberg, Ludwig ’07

Conformal invariance −→

∆θq =

π

θ∆sq

fθ(αθq) =

π

θ[fs(α

sq)− 1] ,

αθq − 2 =π

θ[αsq − 2]

Power-law random banded matrix model (PRBM)

Anderson transition: dimensionality dependence:

d = 2 + ε: weak disorder/coupling dÀ 1: strong disorder/coupling

Evolution from weak to strong coupling – ?

PRBM ADM, Fyodorov, Dittes, Quezada, Seligman ’96

N ×N random matrix H = H† 〈|Hij|2〉 =1

1 + |i− j|2/b2

←→ 1D model with 1/r long range hopping 0 < b <∞ parameter

Critical for any b −→ family of critical theories!

bÀ 1 analogous to d = 2 + ε b¿ 1 analogous to dÀ 1 (?)

Analytics: bÀ 1: σ-model RG

b¿ 1: real space RG

Numerics: efficient in a broad range of bEvers, ADM ’01

Weak multifractality, bÀ 1

supermatrix σ-model

S[Q] =πρβ

4Str

[

πρ∑

rr′Jrr′Q(r)Q(r′)− iω

∑

r

Q(r)Λ

]

.

In momentum (k) space and in the low-k limit:

S[Q] = β Str

[

−1

t

∫

dk

2π|k|QkQ−k −

iπρω

4Q0Λ

]

DOS ρ(E) = (1/2π2b)(4πb− E2)1/2 , |E| < 2√πb

coupling constant 1/t = (π/4)(πρ)2b2 = (b/4)(1− E2/4πb)

−→ weak multifractality

τq ' (q − 1)(1− qt/8πβ) , q ¿ 8πβ/t

E = 0 , β = 1 −→ ∆q =1

2πbq(1− q)

Multifractality in PRBM model: analytics vs numerics

0 0.5 1 1.5α

−1.5

−0.5

0.5

1.5

f(α)

b=4.0b=1b=0.25b=0.01parabola b=4.0parabola b=1.0f0(α*2b)/2b

10−2

10−1

100

101

b

0.0

0.2

0.4

0.6

0.8

1.0

D2

numerics: b = 4, 1, 0.25, 0.01

analytics: bÀ 1 (σ–model RG), b¿ 1 (real-space RG)

Scale-invariant IPR distribution

IPR variance

var(Pq)/〈Pq〉2 = q2(q − 1)2/24β2b2 , q ¿ q+(b) ≡ (2βπb)1/2

IPR distribution function for Pq/〈Pq〉 − 1¿ 1:

P(P ) = e−P−C exp(−e−P−C) , P =

[

Pq

〈Pq〉− 1

]

2πβb

q(q − 1)

C ' 0.5772 – Euler constant

Pq/〈Pq〉 − 1À 1: power-law tail

P(Pq) ∼ (Pq/〈Pq〉)−1−xq , xq = 2πβb/q2 , q2 < 2πβb

Scale-invariant IPR distribution (cont’d)

−7 −6 −5 −4 −3ln P2

10−2

10−1

100

Dis

trib

utio

n (ln

P2)

256512102420484096

−5 0 5 10 15P

10−3

10−2

10−1

100

101

P(P

)

10 10010

−6

10−5

10−4

10−3

~

~Distribution P(lnP2)for b = 1 and L =256, 512, 1024, 2048, 4096

Distribution P(Pq) at b = 4for q = 2 (◦), 4 (¤), and 6 (¦)Solid line — analytical resultfor q ¿ q+(b) = (8π)1/2 ' 5.

Inset: Power-law asymptotics of P(P4).

Power-law exponent: numerically x4 = 1.7,analytically (bÀ 1): x4 = π/2

Strong multifractality, b¿ 1

Real-space RG:

• start with diagonal part of H: localized states with energies Ei = Hii

• include into consideration Hij with |i− j| = 1

Most of them irrelevant, since |Hij| ∼ b¿ 1, while |Ei − Ej| ∼ 1

Only with a probability ∼ b is |Ei − Ej| ∼ b−→ two states strongly mixed (“resonance”) −→ two-level problem

Htwo−level =

(

Ei VV Ej

)

; V = Hij

New eigenfunctions and eigenenergies:

ψ(+) =

(

cos θsin θ

)

; ψ(−) =

(

− sin θcos θ

)

E± = (Ei + Ej)/2± |V |√

1 + τ 2

tan θ = −τ +√

1 + τ 2 and τ = (Ei − Ej)/2V

• include into consideration Hij with |i− j| = 2

• . . .

Strong multifractality, b¿ 1 (cont’d)

Evolution equation for IPR distribution (“kinetic eq.” in “time” t = ln r):

∂

∂ ln rf(Pq, r) =

2b

π

∫ π/2

0

dθ

sin2 θ cos2 θ

× [−f(Pq, r) +

∫

dP (1)q dP (2)

q f(P (1)q , r)f(P (2)

q , r)

× δ(Pq − P (1)q cos2q θ − P (2)

q sin2q θ)]

−→ evolution equation for 〈Pq〉: ∂〈Pq〉/∂ ln r = −2bT (q)〈Pq〉 with

T (q) =1

π

∫ π/2

0

dθ

sin2 θ cos2 θ(1− cos2q θ − sin2q θ) =

2√π

Γ(q − 1/2)

Γ(q − 1)

−→ multifractality 〈Pq〉 ∼ L−τq , τq = 2bT (q)

This is applicable for q>1/2

For q<1/2 resonance approximation breaks down; use ∆q = ∆1−q

Strong multifractality, b¿ 1 (cont’d)

T (q) asymptotics:

T (q) ' −1/[π(q − 1/2)] , q → 1/2 ;

T (q) ' (2/√π)q1/2 , q À 1

Singularity spectrum:

f(α) = 2bF (A) ; A = α/2b , F (A)– Legendre transform of T (q)

F (A) asymptotics:F (A) ' −1/πA , A→ 0 ;

F (A) ' A/2 , A→∞

0 2 4 6 8q

-2

0

2

Tq

0 1 2 3 4A

-2

0

2

4

F(A

)

a)

0.0 0.5 1.0 1.5 2.0αq

-0.8

-0.4

0.0

0.4

0.8

f(α q)

0 5 10q0

1

2

τ qb)

0 0.5 1 1.5 2α

−0.5

0

0.5

1

f(α)

Multifractality in PRBM model: analytics vs numerics

0 0.5 1 1.5α

−1.5

−0.5

0.5

1.5

f(α)

b=4.0b=1b=0.25b=0.01parabola b=4.0parabola b=1.0f0(α*2b)/2b

10−2

10−1

100

101

b

0.0

0.2

0.4

0.6

0.8

1.0

D2

numerics: b = 4, 1, 0.25, 0.01

analytics: bÀ 1 (σ–model RG), b¿ 1 (real-space RG)

Critical level statistics

Two-level correlation function:

R(c)2 (s) = 〈ρ〉−2〈ρ(E − ω/2)ρ(E + ω/2)〉 − 1 ; ρ(E) = V −1Tr δ(E − H)

s = ω/∆, ∆ = 1/〈ρ〉V – mean level spacing

Level number variance:

〈δN(E)2〉 =

∫ 〈N(E)〉

−〈N(E)〉ds(〈N(E)〉 − |s|)R(c)

2 (s)

Spectral compressibility χ : 〈δN 2〉 ' χ〈N〉

unitary symmetry (β = 2):

RMT: R(c)2 (s)=δ(s)− sin2(πs)/(πs)2, χ = 0

Poisson: R(c)2 (s)=δ(s), χ = 1

Criticality: intermediate scale-invariant statistics, 0 < χ < 1

Critical level statistics in PRBM ensemble

• bÀ 1 −→ σ-model −→

R(c)2 (s)=δ(s)−sin2(πs)

(πs)2

(πs/4b)2

sinh2(πs/4b)(β = 2) , χ ' 1/2πβb

• b¿ 1 −→ real-space RG −→R

(c)2 (s) = δ(s)− erfc(|s|/2

√πb) , χ ' 1− 4b , (β = 1)

R(c)2 (s) = δ(s)− exp(−s2/2πb2) , χ ' 1− π

√2 b , (β = 2)

0 0.2 0.4 0.6 0.8 1s

0

0.2

0.4

0.6

0.8

1

R2(s

)

a)

10−2

10−1

100

101

b

0.0

0.2

0.4

0.6

0.8

1.0

χ

Disordered electronic systems: Symmetry classification

Altland, Zirnbauer ’97

Conventional (Wigner-Dyson) classes

T spin rot. chiral p-h symbol

GOE + + − − AIGUE − +/− − − AGSE + − − − AII

Chiral classes

T spin rot. chiral p-h symbol

ChOE + + + − BDIChUE − +/− + − AIIIChSE + − + − CII

H =

(

0 tt† 0

)

Bogoliubov-de Gennes classes

T spin rot. chiral p-h symbol

+ + − + CI− + − + C+ − − + DIII− − − + D

H =

(

h ∆

−∆∗ −hT

)

Mechanisms of Anderson criticality in 2D

“Common wisdom”: all states are localized in 2D

In fact, in 9 out of 10 symmetry classes the system can escape localization!

−→ variety of critical points

Mechanisms of delocalization & criticality in 2D:

• broken spin-rotation invariance −→ antilocalization, metallic phase, MIT

classes AII, D, DIII

• topological term π2(M) = Z (quantum-Hall-type)

classes A, C, D : IQHE, SQHE, TQHE

• topological term π2(M) = Z2

classes AII, CII

• chiral classes: vanishing β-function, line of fixed points

classes AIII, BDI, CII

• Wess-Zumino term (random Dirac fermions, related to chiral anomaly)

classes AIII, CI, DIII

Multifractality at the Quantum Hall critical point

Evers, Mildenberger, ADM ’01

important for identification of the CFT of the Quantum Hall critical point

0.5 1.0 1.5 2.0 2.5α

−1.0

−0.5

0.0

0.5

1.0

1.5

2.0

f (α)

0.8 1.2 1.6 2.0 2.40.0

0.5

1.0

1.5

2.0

f(α)

L=16L=128L=1024

0 1 2q

0.24

0.26

0.28

0.30

∆ q/q(

1−q)

QHE: anomalous dimensions

−→ spectrum is parabolic with a high (1%) accuracy:

f(α) ' 2− (α− α0)2

4(α0 − 2), ∆q ' (α0−2)q(1−q) with α0−2 = 0.262±0.003

recent data, still higher accuracy (unpublished): non-parabolicity is for real!

Multifractal wave functions at the Quantum Hall transition

Spin quantum Hall effect

• disordered d-wave superconductor (class C):

charge not conserved but spin conserved

• time-reversal invariance broken:

• dx2−y2 + idxy order parameter

• strong magnetic field

• Haldane-Rezayi d-wave paired state of composite fermions at ν = 1/2

−→ SQH plateau transition: spin Hall conductivity quantized

jZx = σsxy

(

−dBz(y)

dy

)

Model: SU(2) modification of the Chalker-Coddington network

Kagalovsky, Horovitz, Avishai, Chalker ’99 ; Senthil, Marston, Fisher ’99

Spin quantum Hall effect (cont’d)

Similar to IQH transition but:

• DoS critical ρ(E) ∝ Eµ

• mapping to percolation: analytical evaluation of

• DOS exponent µ = 1/7

• localization length exponent ν = 4/3

• lowest multifractal exponents: ∆2 = −1/4, ∆3 = −3/4

• numerics: analytics confirmed

multifractality spectrum: ∆q, f(α) not parabolic

10−1

100

101

E/δ

0.1

1.0

ρ(E

) L1/

4

0 1 2 3 4E/δ

0

1

2

ρ(E

) L1/

4

0 1 2 3 4q

0.12

0.13

0.14

∆ q / q

(1−

q)

1 1.5 2α−0.5

0

0.5

1

1.5

2

f(α)

2.08 2.1 2.12 2.141.996

1.998

2

Gruzberg, Ludwig, Read ’99; Beamond, Cardy, Chalker ’02; Evers, Mildenberger, ADM ’03