Post on 18-Jun-2020
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