Intermediate spectral statistics Eugene Bogomolny University - - PowerPoint PPT Presentation
Inroduction Pseudo-integrable models Pseudo-integrable map Random Lax matrices Conclusion Intermediate spectral statistics Eugene Bogomolny University Paris-Sud, CNRS Laboratoire de Physique Th eorique et Mod` eles Statistiques, Orsay
Inroduction Pseudo-integrable models Pseudo-integrable map Random Lax matrices Conclusion
Collaborators : C. Schmit, O. Giraud, R. Dubertrand Conference on Frontiers of Nanoscience, Trieste 2015
Eugene Bogomolny, LPTMS, Orsay , France Intermediate statistics
Inroduction Pseudo-integrable models Pseudo-integrable map Random Lax matrices Conclusion
1 Inroduction 2 Pseudo-integrable models 3 Pseudo-integrable map 4 Random Lax matrices 5 Conclusion Eugene Bogomolny, LPTMS, Orsay , France Intermediate statistics
Inroduction Pseudo-integrable models Pseudo-integrable map Random Lax matrices Conclusion
Integrable systems (localized states) : spectral statistics = Poisson statistics Berry , Tabor (1977)
1 2 3
s
0.2 0.4 0.6 0.8 1
p(s)
p(s)=exp(−s)
Chaotic systems (extended states) : spectral statistics = random matrix statistics Bohigas, Giannoni, Schmit (1984)
1 2 3
s
0.2 0.4 0.6 0.8 1
p(s)
p(s)=π/2 s exp(−π s
2/4)
Eugene Bogomolny, LPTMS, Orsay , France Intermediate statistics
Inroduction Pseudo-integrable models Pseudo-integrable map Random Lax matrices Conclusion
H =
εia†
i ai −
a†
j ai
εi = i.i.d. ∈ [−W /2, W /2] Mobility edge : Ec(W ) W > Wc ≈ 16.5 → all states are localized
Ec −Ec
density
localized states localized states delocalized states
|E| > Ec. States are localized. Poisson statistics of eigenvalues |E| < Ec. States are delocalized. Random matrix statistics |E| = Ec. States are neither localized or delocalized. Fractal wave functions. Intermediate type of spectral statistics. Shklovskii et al. (1993) Spectral characteristics of 3-d Anderson model at metal-insulator transition
1 2 3 4
s
0.5 1
Nearest−neighbor distribution
Insulator, W=100 Metal, W=5 Critical, W=16.5
1 2 3 4 5
L
1 2 3 4 5
Number variance Insulator, W=100
Critical, W=16.5 Metal, W=5
Eugene Bogomolny, LPTMS, Orsay , France Intermediate statistics
Inroduction Pseudo-integrable models Pseudo-integrable map Random Lax matrices Conclusion
Level repulsion at small distances (head) as for the RMT p(s) → 0 when s → 0 Exponential decrease of p(s) at large distances (tail) as for the Poisson p(s) ∼ e−as when s → ∞
Eugene Bogomolny, LPTMS, Orsay , France Intermediate statistics
Inroduction Pseudo-integrable models Pseudo-integrable map Random Lax matrices Conclusion
Level repulsion at small distances (head) as for the RMT p(s) → 0 when s → 0 Exponential decrease of p(s) at large distances (tail) as for the Poisson p(s) ∼ e−as when s → ∞ Merman
Eugene Bogomolny, LPTMS, Orsay , France Intermediate statistics
Inroduction Pseudo-integrable models Pseudo-integrable map Random Lax matrices Conclusion
Level repulsion at small distances (head) as for the RMT p(s) → 0 when s → 0 Exponential decrease of p(s) at large distances (tail) as for the Poisson p(s) ∼ e−as when s → ∞ Merman Linear asymptotics of the number variance Σ2(L) ≡ (n(L) − ¯ n(L))2 → χL when L → ∞ χ = spectral compressibility χ = 1 for Poisson, χ = 0 for the RMT
L n(L)
Multi-fractal character of eigenfunctions |Ψ|2q → V −(q−1)Dq when V → ∞, V = system size Dq = fractal dimensions Dq = 0 for the Poisson, Dq = 1 for the RMT
Eugene Bogomolny, LPTMS, Orsay , France Intermediate statistics
Inroduction Pseudo-integrable models Pseudo-integrable map Random Lax matrices Conclusion
Critical random matrices, Levitov (1990), Altshuler & Levitov (1997) Mij ∼ εj δij +
g |i−j|α ,
α < 1 → RMT, α > 1 → Posson, α = 1 → critical Critical power law banded random matrices, Mirlin et al. (1996) Hij = i.i.d. Gaussian variables, (β = 1, 2) Hij = 0, |Hij |2 =
b2
−1 i = j , |Hii|2 = β−1 b → ∞ = ⇒ RMT, b → 0 = ⇒ Poisson Main results Perturbation series, Mirlin, Evers (2000) b ≫ 1 : Dq = 1 − q/(2πβb), χ = 1/(2πβb), Dq = 1 − qχ b ≪ 1 : (cβ = 1 for β = 1, cβ = π/ √ 8 for β = 2) Dq = 4bcβ Γ(q − 1/2) √πΓ(q) , χ = 1 − 4bcβ, Dq = Γ(q − 1/2) √πΓ(q) (1 − χ) Symmetry, Mirlin et al. (2006) ∆q ≡ (Dq − 1)(q − 1), ∆q = ∆1−q
Eugene Bogomolny, LPTMS, Orsay , France Intermediate statistics
Inroduction Pseudo-integrable models Pseudo-integrable map Random Lax matrices Conclusion
Q : Do dynamical models with intermediate statistics exist ? R : Yes, pseudo-integrable models
Finite genus 2-dim surfaces g = 1 + N 2
mi − 1 ni , N = the least common multiple of ni Triangles [π/4, π/4, π/2] and [π/6, π/3, π/2] are integrable − → g = 1 (torus) Triangles [π/5, 3π/10, π/2] and [π/8, 3π/8, π/2] are not − → g = 2
Eugene Bogomolny, LPTMS, Orsay , France Intermediate statistics
Inroduction Pseudo-integrable models Pseudo-integrable map Random Lax matrices Conclusion
Unfolding of π/8 right triangle
Interval-exchange map I1 = [0, 1], I2 = [1, 2], I3 = [2, 3], I4 = [3, 4] [0, 7] = I4, [7, 6] = I3, [6, 5] = I2, [5, 4] = I1 I1, I2, I3, I4 − → I4, I3, I2, I1 Neither integrable nor chaotic
Eugene Bogomolny, LPTMS, Orsay , France Intermediate statistics
Inroduction Pseudo-integrable models Pseudo-integrable map Random Lax matrices Conclusion
1 2 3 4
s
0.0 0.6 1.2 1.8
N(s)
1 2 3
s
0.0 0.5 1.0
R(s)
2 4 s −0.10 −0.05 0.00 0.05 N(s)−Nsp(s)
Left : Cumulative nearest-neighbour distribution, N (s) = s
0 p(t)dt, for 10 000
levels for π/5 right triangle. Each curve contains 2500 consecutive levels · · · Poisson : pp(s) = e−s, – – – RMT : pWigner(s) = π/2se−πs2/4 —– Semi-Poisson : psp(s) = 4se−2s, R2(s) = 1 − e−4s Right : Difference between N (s) for π/n right triangles with n = 5, 7, . . . , 30 and the Semi-Poisson formula Nsp(s) = 1 − (2s + 1)e−2s. (5000-20000 levels). Closest lines correspond to n = 5, 8, 10, 12
Eugene Bogomolny, LPTMS, Orsay , France Intermediate statistics
Inroduction Pseudo-integrable models Pseudo-integrable map Random Lax matrices Conclusion
π/n right triangle
π n
1 2 3 t 0.5 1 1.5 K(t)χ ≡ K(0) = n + ǫ(n) 3(n − 2) , ǫ(n) = when n is odd 2 when n is even but not divisible by 3 6 when n is divisible by 6 Sum over all periodic orbit by using the Veech group. Tedious calculations Rectangular billiard with a flux line
Aharonov−Bohm flux line
Aharonov-Bohm flux line Aφ = α/r at point x0, y0 Ψn(r, φ) = 0 on a rectangle a, b χ ≡ K(0) = 1 − 4α(1 − α) + 6αη η = explicit function of e1 = x0/a and e2 = y0/b, For irrational e1, e2, η = 1/6
Eugene Bogomolny, LPTMS, Orsay , France Intermediate statistics
Inroduction Pseudo-integrable models Pseudo-integrable map Random Lax matrices Conclusion
Wedge, γ = α/π
α θi θf
ϕ √ kr ∼ 1 D(θf , θi) = 2 γ sin π γ
γ − cos θf + θi γ −1 −
γ − cos θf − θi γ −1 Flux line D(θf , θi) = − i sin πα 2 cos[(θf − θi)/2] ei(θf −θi )/2
α eiπα e πα i −
Formation of wave propagating in periodic orbit channels
δϕ
α
ϕ reflection small angle reflection large angle specular direction 2π−2α
Dominant effect : mirror reflection ϕ ≈ p/k < 1 Ψ(x, y) ∼ sin py eiqx p = π w n, q = π Lm E = p2 + q2
ϕ w
Eugene Bogomolny, LPTMS, Orsay , France Intermediate statistics
Inroduction Pseudo-integrable models Pseudo-integrable map Random Lax matrices Conclusion
m = 347, n = 1 ; E347,1 = 10041.87 Eexact = 10041.41 m = 228, n = 1 ; E228,1 = 10106.31 Eexact = 10106.20 Fourier expansion Ψ(x, y) =
CmnΨmn(x, y), Ψ(e)
mn(x, y) = cos π
a (m − 1 2 )x sin π b ny
20 40 60 80 100 120 140 160 0 10 20 30 40 50 60 70 80 90
Eexact = 10041.41
20 40 60 80 100 120 140 160 0 20 40 60 80 100 120
Eexact = 10106.20
Eugene Bogomolny, LPTMS, Orsay , France Intermediate statistics
Inroduction Pseudo-integrable models Pseudo-integrable map Random Lax matrices Conclusion
10000 10100 10200 50 100 4000 4200 4400 4600 4800 20 40 60 80 100 1000 1200 1400 1600 1800 20 40 60 10000 10100 10200 10300 10400 50 100 4000 4200 4400 4600 4800 20 40 60 80 100 1000 1200 1400 1600 1800 20 40 60
Barrier billiard Stadium
50 100 k 20 40 60 80 100 PR(k) Stadium Barrier
Stadium fit : R2(k) = 3
4 k + 1
Barrier billiard fit : R2(k) = 2.55 √ k
5000
E
2000
R3
60
R2
Participation ratios R2 (top) and R3 (bottom) Fit : R2 ≈ 2.52 √ k, R3 ≈ 4.7k. Fractal dimensions : D2 ≈ D3 ≈ .5
Eugene Bogomolny, LPTMS, Orsay , France Intermediate statistics
Inroduction Pseudo-integrable models Pseudo-integrable map Random Lax matrices Conclusion
p x
→
Φ0
x + f (p)
p x
→
ρα
p + α x
p x
→
Φα
x + f (p + α)
→ pk = p + km/n − → Φn
α = pseudo-integrable map
p x
→
Φn
α
x + C
C =
N
f (p + j α), x − → x + C mod 1
1 1
1−C C
Quantum map ≈ a unitary matrix whose saddle points = classical map Q′|U (Φα)|Q = 1 N
N−1
e2πiSk (Q′,Q), Sk(Q′, Q) = −N Φ k N
N (Q′−Q)+αQ Q, Q′ = 1, . . . , N , Φ′(p) = f (p) - Giraud, Marklof, O’Keefe (2004) x = Q/N , x ′ = Q′/N , p′ = ∂Q′Sk(Q′, Q), p = −∂QSk(Q′, Q) − → x ′ = x + f (p′) ”Classical trajectory” − → ∂kSk(Q′, Q) = 0 − → p′ = p + α,
Eugene Bogomolny, LPTMS, Orsay , France Intermediate statistics
Inroduction Pseudo-integrable models Pseudo-integrable map Random Lax matrices Conclusion
Unitary N × N matrix Mkp = eiΦk 1 − e2πiαN N [1 − e2πi(k−p+αN)/N ] , k, p = 1, . . . , N Two cases Non-symmetric (analog of GUE) : Φk are i.i.d. random variables with uniform distribution between 0 and 2π. With ‘time reversal symmetry’ (analog of GOE) : A half of coefficients is
Irrational α = √ 5/2, N = 201
1 2 3 s 0.5 1 p(s)
Solid lines : the Wigner sur- mises p(s) = π 2 se−π2s2/4 = ⇒ GOE p(s) = 32 π2 s2e−4s2/π2 = ⇒ GUE
Eugene Bogomolny, LPTMS, Orsay , France Intermediate statistics
Inroduction Pseudo-integrable models Pseudo-integrable map Random Lax matrices Conclusion
For rational α = m/q and mN ≡ ±1 (mod q) spectral statistics of the main matrix is given by the semi-Poisson statistics with β = q − 1 for non-symmetric matrices
q−2 2
for symmetric matrices Semi-Poisson statistics pβ(E1, E2, . . . , EN ) ∼
N−1
|En+1 − En|β, E1 < E2 <, . . . , < EN Interaction only with nearest-neighbour levels Nearest-neighbour spacing distribution p(s) = Aβsβe−(β+1)s, Aβ = (β + 1)β+1/Γ(β + 1) Two-point form-factor : K(t) = 1 + 2Re
−1 Level compressibility, (Σ2(L) → χL) : χ = K(0) = (β + 1)−1 The simplest case : β = 1 p(s) = 4se−2s, R2(s) = 1 − e−4s, χ = 1/2
Eugene Bogomolny, LPTMS, Orsay , France Intermediate statistics
Inroduction Pseudo-integrable models Pseudo-integrable map Random Lax matrices Conclusion
Non-symmetric matrices α = 1/3, 1/6, 1/9
1 2 3 s 0.5 1 1.5 p(s)
α = 1/3, p(s) = A2s2e−3s α = 1/6, p(s) = A5s5e−6s α = 1/9, p(s) = A8s8e−9s Symmetric matrices, α = 1/2, 1/3, 1/5, 1/7
2 4 6 s 0.5 1 p(s)
α = 1/2, p(s) = e−s α = 1/3, p(s) = A1/2s1/2e−3/2s α = 1/5, p(s) = A3/2s3/2e−5/2s α = 1/7, p(s) = A5/2s5/2e−7/2s
Eugene Bogomolny, LPTMS, Orsay , France Intermediate statistics
Inroduction Pseudo-integrable models Pseudo-integrable map Random Lax matrices Conclusion
Nearest-neighbour distribution
1 2 3 s 0.5 1 1.5 2 p(s)
N = 801, p(s) = A19s19e−20s N = 809, p(s) ≈ 32 π2 s2e−4s2/π2 Two-point correlation function N = 801
1 2 3 4 5 s 0.5 1 1.5 2 R2(s)
R2(s) = s−20s
19
exp
20
α = m/q and mN ≡ −k mod q. Exact but tedious calculations Correlation functions are calculated from transfer matrix of dimension C k−1
q−2
χ = 1/q for non-symmetric, χ = 2/q for symmetric cases
Eugene Bogomolny, LPTMS, Orsay , France Intermediate statistics
Inroduction Pseudo-integrable models Pseudo-integrable map Random Lax matrices Conclusion
Fractal dimensions |Ψ|2q → N −(q−1)Dq α = 1/3 (black) α = 1/5 (red) α = 1/9 (blue) Symmetry relation ∆q ≡ (Dq − 1)(q − 1) ∆q (solid), ∆1−q (dashed) Eigenfunctions of quantum map are multifractal Symmetry relation, ∆q = ∆1−q, is not fulfilled
Eugene Bogomolny, LPTMS, Orsay , France Intermediate statistics
Inroduction Pseudo-integrable models Pseudo-integrable map Random Lax matrices Conclusion
Random matrix Lkr = pr δkr + ig 1 − δkr qk − qr pj , qj = random variables with ’natural’ measure density P( p, q ) ∼ e
−α
k q2 k = e
−α
k +g2 i=j 1 (qi −qj )2
k q2 k
Lkr = the Lax matrix of the Calogero-Moser model : ˙ L = [L, M ] Calogero-Moser Hamiltonian : H (p, q) =
j 1 2 p2 j + g2 j<k 1 (qj −qk )2
Eigenvalues and eigenfunctions
m u∗ k (m)ur (m) = δkr
New matrix : Qmn =
u∗
k (m)qk uk(n) = wmδmn − ig 1 − δmn
λm − λn . wm=angle variables, λm= action variables, Ruijsenaars (1988) Canonical transformation : d pd q = d λd w
Eugene Bogomolny, LPTMS, Orsay , France Intermediate statistics
Inroduction Pseudo-integrable models Pseudo-integrable map Random Lax matrices Conclusion
dL ∼ e−α
k q2 k dpdq
TrL†L =
λ2
m,
q2
k = TrQ†Q =
w2
m+
g2 (λm − λn)2 , d pd q = d λd w dL ∼ e
−α
m λ2 m−β
m+g2 m=n 1 (λm −λn )2
Exact joint eigenvalue distribution without explicit group invariance P( λ ) ∼ e
−α
m λ2 m−βg2 m=n 1 (λm −λn )2
pr = i.i.d. variables k, r = 1, . . . , N Lkr = pr δkr + ig 1−δkr
k−r
g = 0.1, 0.5, 1, 2 Wigner-type anzatz : p(s) = Ae−B/s2−Cs
0.5 1 1.5 2 2.5 3 s 1 2 3 P(s)
0.5 1 1.5 2 2.5 3
0.05 Eugene Bogomolny, LPTMS, Orsay , France Intermediate statistics
Inroduction Pseudo-integrable models Pseudo-integrable map Random Lax matrices Conclusion
Calogero-Moser models
Rational : H (p, q) = N
j=1 1 2 p2 j + g2 1≤j<k≤N 1 (qj −qk )2
Hyperbolic : H (p, q) = N
j=1 1 2 p2 j + 1 4 g2µ2 1≤j<k≤N 1 sinh2( µ 2 (qj −qk ))
Trigonometric : H (p, q) = N
j=1 1 2 p2 j + 1 4 g2µ2 1≤j<k≤N 1 sin2( µ 2 (qj −qk )/2)
Ruijsenaars-Schneider model H (p, q) = N
j=1 cos(pj ) k=j
sin2 πa sin2 µ
2 (qj −qk )
1/2 Lax matrices (simplified) Rational : Lkr = prδkr + ig 1 − δkr k − r Hyperbolic : Lkr = pr δkr + ig µ(1 − δkr ) 2 sinh(µ(k − r)/2) Trigonometric : Lkr = pr δkr + ig µ(1 − δkr ) 2 sin(µ(k − r)/2) RS : Lkr = eiΦk
1−e2πia N 1−e2πi(k−r+a)/N
Eugene Bogomolny, LPTMS, Orsay , France Intermediate statistics
Inroduction Pseudo-integrable models Pseudo-integrable map Random Lax matrices Conclusion
Hyperbolic CM Lkr = pr δkr + ig µ(1 − δkr) 2 sinh(µ(k − r)/2) µ = 4π/N , g = 0.05, 0.25, 0.5, 1 Wigner-type anzatz : P(n, s) = asde−b/s−cs Trigonometric CM Lkr = pr δkr + ig µ(1 − δkr ) 2 sin(µ(k − r)/2) µ = 4π/N , g = 0.05, 0.25, 0.5, 1 P(n, s) = 0, 0 < s < nb P(n, s) = (s − nb)n (n − 1)!(1 − b)n e−(s−nb)/(1−b), s > nb
Eugene Bogomolny, LPTMS, Orsay , France Intermediate statistics
Inroduction Pseudo-integrable models Pseudo-integrable map Random Lax matrices Conclusion
Lkr = eiΦk
1−e2πia N
Φk = i.i.d.r.v. uniform between 0, 2π Exact results for all nearest-neighbour distributions Strongly depends on integer part of a 0 < a < 1 : Shifted Poisson distributions, P(n, s) = 0, 0 < s < na P(n, s) = (s − na)n (n − 1)!(1 − a)n e−(s−na)/(1−a), s > na χ = (1 − a)2 a = 4/3 : p(s) = 81 64 s2, 0 < s < 4/3 p(2, s) =
2 + 27 16s − 81 512 s3 e3s/4−1, 4/3 < s < 8/3 p(3, s) = 3 4 − 81 32 s + 81 512 s3 e3s/4−1 + 81 64 s2 4/3 < s < 8/3
4 + 27 32 s − 81 512 s3 e3s/4−1 + 9e3s/2−4 8/3 < s < 4 χ = 4/9
Eugene Bogomolny, LPTMS, Orsay , France Intermediate statistics
Inroduction Pseudo-integrable models Pseudo-integrable map Random Lax matrices Conclusion
a = 1
2 ,
a = 6
5 ,
a = 4
3,
a = 9
4
Eugene Bogomolny, LPTMS, Orsay , France Intermediate statistics
Inroduction Pseudo-integrable models Pseudo-integrable map Random Lax matrices Conclusion
Rational CM model, g = 0.005, 0.025, 0.05, 0.15, 0.25, 0.4 RS model, a = 0.01, 0.05, 0.1, 0.2, 0.3, 0.5, 0.7, 0.9
Eugene Bogomolny, LPTMS, Orsay , France Intermediate statistics
Inroduction Pseudo-integrable models Pseudo-integrable map Random Lax matrices Conclusion
Fractal dimensions in first order of perturbations series for CrBRME and RSE CrBRME RSE Weak multifractality 1/b ≪ 1 |a − k| ≪ 1 Dq = 1 − q
1 2π β b
Dq = 1 − q (a−k)2
k2
χ =
1 2π β b
χ = (a−k)2
k2
Strong multifractality, q > 1/2, c1 = 1, c2 = π/ √ 8 b ≪ 1 |a| ≪ 1 Dq = 4bcβ
Γ
2
Dq = 2a
Γ
2
χ = 1 − 4bcβ χ = 1 − 2a CrBRME RSE Strong multifractality, q < 1/2, c1 = 1, c2 = π/ √ 8 b ≪ 1 |a| ≪ 1 Dq = 2q−1
q−1 − 4bcβ Γ
1 2 −q
√πΓ(1−q)
Dq = 2q−1
q−1 − 2a Γ
1 2 −q
√π Γ(1−q)
Symmetry ∆q = ∆1−q is valid in the first order and well confirmed by numerics
Eugene Bogomolny, LPTMS, Orsay , France Intermediate statistics
Inroduction Pseudo-integrable models Pseudo-integrable map Random Lax matrices Conclusion
N →∞ D1 ln N Critical banded random matrices, χ (black circles), 1 − D1 (red error bars)
0.1 0.15 0.2 0.25 0.3 b 0.3 0.4 0.5 0.6 χ, 1-D1
0.5 1 1.5 2 b 0.2 0.4 0.6 0.8 1 χ, 1-D1
β = 2
1 2 3 4 5 b 0.2 0.4 0.6 0.8 1 χ, 1-D1
0.1 0.2 0.3 0.4 0.5 b 0.3 0.4 0.5 0.6 0.7 χ, 1-D1
β = 1 Ruijsenaars-Schneider ensemble Ultrametric matrices
0.5 1 1.5 2 2.5 3 a 0.2 0.4 0.6 0.8 1 χ, 1-D1
1 1.5 2 2.5 3 0.02 0.04 χ, 1-D1
1 2 3 4 5 J/W 0.2 0.4 0.6 0.8 1 χ, 1-D1
0.2 0.4 0.6 0.8 J/W 0.2 0.3 0.4 0.5 χ, 1-D1
Eugene Bogomolny, LPTMS, Orsay , France Intermediate statistics
Inroduction Pseudo-integrable models Pseudo-integrable map Random Lax matrices Conclusion
Physical problems giving rise to intermediate statistics of eigenvalues
Rich variety of intermediate statistics
→ one large diagonal + (i − j)−1 falloff
Absence of universality Lax matrices of integrable classical systems give new soluble ensembles of random matrices with intermediate statistics Eigenfunctions of models with intermediate statistics are multifractal
diagonal but is ruled out (numerically) for models related with pseudo-integrable maps with a few large diagonals
Conjecture : χ = 1 − D1/d New perspectives for intermediate statistics
Eugene Bogomolny, LPTMS, Orsay , France Intermediate statistics