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 France 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 Outlook 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 Well accepted conjectures Integrable systems (localized states) : spectral statistics = Poisson statistics Berry , Tabor (1977) 1 0.8 p(s)=exp(−s) 0.6 (∆+Ε)Ψ=0 p(s) 0.4 0.2 0 0 1 2 3 s Chaotic systems (extended states) : spectral statistics = random matrix statistics Bohigas, Giannoni, Schmit (1984) 1 p(s)= π /2 s exp(− π s 2 /4) 0.8 0.6 (∆+Ε)Ψ=0 p(s) 0.4 0.2 0 0 1 2 3 s Eugene Bogomolny, LPTMS, Orsay , France Intermediate statistics

Inroduction Pseudo-integrable models Pseudo-integrable map Random Lax matrices Conclusion 3-d Anderson model ε i a † a † � � H = i a i − j a i delocalized states i adjacent ( j , i ) density localized states localized states ε i = i.i.d. ∈ [ − W / 2 , W / 2] Mobility edge : E c ( W ) 0 W > W c ≈ 16 . 5 → all states are localized − E c E c | E | > E c . States are localized. Poisson statistics of eigenvalues | E | < E c . States are delocalized. Random matrix statistics | E | = E c . 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 5 Nearest−neighbor distribution Critical, W=16.5 Insulator, W=100 4 Number variance Metal, W=5 3 Critical, W=16.5 0.5 2 Metal, W=5 1 Insulator, W=100 0 0 0 1 2 3 4 0 1 2 3 4 5 s L Eugene Bogomolny, LPTMS, Orsay , France Intermediate statistics

Inroduction Pseudo-integrable models Pseudo-integrable map Random Lax matrices Conclusion Characteristic features of intermediate statistics 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 Characteristic features of intermediate statistics 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 Characteristic features of intermediate statistics 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 n(L) Σ 2 ( L ) ≡ � ( n ( L ) − ¯ n ( L )) 2 � → χ L when L → ∞ χ = spectral compressibility L χ = 1 for Poisson, χ = 0 for the RMT Multi-fractal character of eigenfunctions �| Ψ | 2 q � → V − ( q − 1) D q when V → ∞ , V = system size D q = fractal dimensions D q = 0 for the Poisson, D q = 1 for the RMT Eugene Bogomolny, LPTMS, Orsay , France Intermediate statistics

Inroduction Pseudo-integrable models Pseudo-integrable map Random Lax matrices Conclusion Random matrix models with intermediate statistics Critical random matrices, Levitov (1990), Altshuler & Levitov (1997) g M ij ∼ ε j δ ij + | i − j | α , α < 1 → RMT , α > 1 → Posson , α = 1 → critical Critical power law banded random matrices, Mirlin et al. (1996) H ij = i.i.d. Gaussian variables, ( β = 1 , 2) � − 1 1 + ( i − j ) 2 � �| H ij | 2 � = �| H ii | 2 � = β − 1 � H ij � = 0 , i � = j , b 2 b → ∞ = ⇒ RMT, b → 0 = ⇒ Poisson Main results Perturbation series, Mirlin, Evers (2000) b ≫ 1 : D q = 1 − q / (2 πβ b ) , χ = 1 / (2 πβ b ) , D q = 1 − q χ √ b ≪ 1 : ( c β = 1 for β = 1, c β = π/ 8 for β = 2) Γ( q − 1 / 2) D q = Γ( q − 1 / 2) D q = 4 bc β √ π Γ( q ) , χ = 1 − 4 bc β , √ π Γ( q ) (1 − χ ) Symmetry, Mirlin et al. (2006) ∆ q ≡ ( D q − 1)( q − 1) , ∆ q = ∆ 1 − q Eugene Bogomolny, LPTMS, Orsay , France Intermediate statistics

Inroduction Pseudo-integrable models Pseudo-integrable map Random Lax matrices Conclusion Pseudo-integrable polygonal billiards Q : Do dynamical models with intermediate statistics exist ? R : Yes, pseudo-integrable models π mi n π i n Finite genus 2-dim surfaces g = 1 + N m i − 1 � , N = the least common multiple of n i 2 n i i Triangles [ π/ 4 , π/ 4 , π/ 2] and [ π/ 6 , π/ 3 , π/ 2] are integrable − → g = 1 (torus) Triangles [ π/ 5 , 3 π/ 10 , π/ 2] and [ π/ 8 , 3 π/ 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 Classical mechanics of pseudo-integrable billiards 4 5 6 1 2 3 0 4 7 6 5 7 Unfolding of π/ 8 right triangle 0 1 2 3 4 Interval-exchange map I 1 = [0 , 1] , I 2 = [1 , 2] , I 3 = [2 , 3] , I 4 = [3 , 4] [0 , 7] = I 4 , [7 , 6] = I 3 , [6 , 5] = I 2 , [5 , 4] = I 1 I 1 , I 2 , I 3 , I 4 − → I 4 , I 3 , I 2 , I 1 Neither integrable nor chaotic Eugene Bogomolny, LPTMS, Orsay , France Intermediate statistics

Inroduction Pseudo-integrable models Pseudo-integrable map Random Lax matrices Conclusion Spectral statistics for π/ n right triangles (numerics) 0.05 1.8 0.00 1.2 N(s)−Nsp(s) N(s) 1.0 −0.05 0.6 R(s) 0.5 0.0 −0.10 0 1 2 3 s 0 2 4 0.0 s 0 1 2 3 4 s � s Left : Cumulative nearest-neighbour distribution, N ( s ) = 0 p ( t )d t , for 10 000 levels for π/ 5 right triangle. Each curve contains 2500 consecutive levels · · · Poisson : p p ( s ) = e − s , – – – RMT : p Wigner ( s ) = π/ 2 s e − π s 2 / 4 —– Semi-Poisson : p sp ( s ) = 4 s e − 2 s , R 2 ( s ) = 1 − e − 4 s Right : Difference between N ( s ) for π/ n right triangles with n = 5 , 7 , . . . , 30 and the Semi-Poisson formula N sp ( s ) = 1 − (2 s + 1)e − 2 s . (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 Calculation of level compressibility π/ n right triangle 1.5 1 K(t) π 0.5 n 0 0 1 2 3 t  0 when n is odd χ ≡ K (0) = n + ǫ ( n )  3( n − 2) , ǫ ( n ) = 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 A φ = α/ r at point x 0 , y 0 Ψ n ( r , φ ) = 0 on a rectangle a , b χ ≡ K (0) = 1 − 4 α (1 − α ) + 6 αη η = explicit function of e 1 = x 0 / a and e 2 = y 0 / b , For irrational e 1 , e 2 , η = 1 / 6 Eugene Bogomolny, LPTMS, Orsay , France Intermediate statistics

Inroduction Pseudo-integrable models Pseudo-integrable map Random Lax matrices Conclusion Strong diffraction Wedge , γ = α/π √ ϕ kr ∼ 1 θ f α θ i D ( θ f , θ i ) = 2 γ sin π �� cos π γ − cos θ f + θ i cos π γ − cos θ f − θ i � − 1 � � − 1 � − γ γ γ Flux line ei πα i sin πα α 2 cos[( θ f − θ i ) / 2] e i( θ f − θ i ) / 2 D ( θ f , θ i ) = − πα − i e Formation of wave propagating in periodic orbit channels ϕ ≈ p / k < 1 specular large angle direction reflection 2π−2α small angle δϕ reflection Ψ( x , y ) ∼ sin py e i qx ϕ α w ϕ p = π w n , q = π Lm Dominant effect : E = p 2 + q 2 mirror reflection Eugene Bogomolny, LPTMS, Orsay , France Intermediate statistics

Inroduction Pseudo-integrable models Pseudo-integrable map Random Lax matrices Conclusion Superscars m = 347, n = 1 ; E 347 , 1 = 10041 . 87 m = 228, n = 1 ; E 228 , 1 = 10106 . 31 E exact = 10106 . 20 E exact = 10041 . 41 Fourier expansion mn ( x , y ) = cos π a ( m − 1 2 ) x sin π Ψ ( e ) � Ψ( x , y ) = C mn Ψ mn ( x , y ) , b ny mn 90 120 80 100 70 60 80 50 0 0 60 20 40 20 40 40 60 30 60 40 80 80 20 100 100 20 120 10 120 140 140 160 0 160 0 E exact = 10041 . 41 E exact = 10106 . 20 Eugene Bogomolny, LPTMS, Orsay , France Intermediate statistics