anderson transitions and wave function multifractality ...anderson transitions and wave function...
TRANSCRIPT
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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/
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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
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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
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Experiment:
Wave function statistics in disordered microwave billiards
Kudrolli, Kidambi, Sridhar, PRL 95
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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
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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
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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. . .
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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
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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
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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
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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
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Multifractal wave functions at the Quantum Hall transition
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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
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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
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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+
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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
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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
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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
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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
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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
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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
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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
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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]
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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
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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)
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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)
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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
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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
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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
• . . .
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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
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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)
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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)
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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
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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
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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
)
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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
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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!
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Multifractal wave functions at the Quantum Hall transition
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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
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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