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SCARS ON GRAPHS
Holger Schanz and Tsampikos Kottos
Nichtlineare Dynamik Universit¨ at G¨
- ttingen
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Deterministic Chaos vs Random Walk on a Graph
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SCARS ON GRAPHS Holger Schanz and Tsampikos Kottos Nichtlineare - - PowerPoint PPT Presentation
SCARS ON GRAPHS Holger Schanz and Tsampikos Kottos Nichtlineare - - PowerPoint PPT Presentation
SCARS ON GRAPHS Holger Schanz and Tsampikos Kottos Nichtlineare Dynamik Universit at G ottingen 1 Deterministic Chaos vs Random Walk on a Graph 1 2 Quantum graphs: 1. Line h 2 d 2 d x 2 ( x ) = E ( x ) 2m h 2 k =
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Quantum graphs:
- 1. Line
− ¯ h2 2m d2 dx2 Ψ(x) = E Ψ(x) k =
- 2mE/¯
h2
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Quantum graphs:
- 2. Vertex
a− = σ a+
- current conservation σσ† = I
- continuity of wavefunction
Neumann b.c.: σ = ρ τ τ τ ρ τ · · · τ τ ρ · · · · · · τ = 2 v ≪ 1 (v → ∞) ρ = 2 v − 1 ∼ −1 (v → ∞)
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Quantum graphs:
- 3. Network
Ud′,d(k) = exp(ikLlm) σ(m)
ln
d = [l → m] d′ = [m → n] interpretation: discrete time evolution |ψ(t) = Ut(k)| ψ(0) classical analogue: Markov chain Md′d = |Ud′d|2
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A complete Graph
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An “ergodic” eigenstate
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A typical eigenstate
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A scar on a graph
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The inverse participation number
2B
- d=1
|ad|2 = 1 I =
2B
- d=1
|ad|4 < 1
1 2B = 1 380
0.0096 ≈
1 104
0.1516 ≈ 1
6 1 2
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IPN from return probability
Heller ’84: scars ⇔ short-time dynamics: d|Ut|d =
- m
d|m e−iǫmt m|d Pd(t)t =
- m,n
|m|d|2 |n|d|2 ei(ǫm−ǫn)tt
- δm,n
Pd(t)t,d = Imm ⇒ average localization of eigenstates ∆ǫ ∼ 2π
2B
⇒ ·t ∼
1 2B
2B
t=1
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Many ways to return ...
eikL1 ρ eikL1 ρ eikL1 τ eikL2 τ eikL3 τ Neumann b.c., large graphs: τ = 2/v → ρ = τ − 1 → −1
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... but only one is important
Kaplan 2001: period-two orbits ⇒ I ∼ v × IRMT The shortest and most stable orbits cause enhanced localization.
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Take-home message
Weak scars = Strong scars
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Which orbits can scar?
j p
D(+)
j,p , D(−) j,p ,
D(+)
j,p ,
D(−)
j,p
perfect scars: 0 = 0 + τ
- d∈D(−)
j,p
eikLd ad vj,p ≥ 2 − δvj,1 (∀j ∈ p) Stability is irrelevant!
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Energies of scars?
j d ^ d p
D(+)
j,p , D(−) j,p ,
D(+)
j,p ,
D(−)
j,p
ad =
- d′∈D(−)
j,p
eikLd′ ad′ (τ
- +δd′ˆ
d [ρ − τ] −1
) ad = −eikLd aˆ
d
ad = +e2ikLd ad kLd = mdπ ∀ d ∈ p ⇒ commensurate bond lengths Ld/Ld′ = md/md′ No perfect scars for generic graphs!
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Perturbation theory for the scar quality
P(ε → 0) ∼ εN−2 |ascar = |a(0)
scar + ε
- m=scar
a(0)
m |ˆ
Φ|a(0)
scar
1 − exp(i[λ(0)
m − λ(0) scar])
|a(0)
m
Scar quality: δp =
d/ ∈p |ad|2
(0 ≤ δ ≤ 1, δp = 0: perfect scar)
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Distribution of scars
P(N)(δ → 0) = C δ(N−3)/2 P(2)(δ → 0) ∼ δ−1/2 P(3)(δ → 0) = C P(4)(δ → 0) ∼ δ+1/2 I ≥ 1/6 − δ/3 cf Berkolaiko et al. (2003): No quantum ergodicity for star graphs.
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Conclusions
- The scar theory of Heller et al. applies to graphs, ...
- ... but it does not describe the scars ...
- ... because strong and weak scarring are unrelated
phenomena.
- A detailed understanding of strong scars was achieved, ...
- ... but the method does not (immediately) generalize to other systems.
- Phys. Rev. Lett. 90 (03) 234101.
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