String-like theory of many-particle Quantum Chaos Boris Gutkin - PowerPoint PPT Presentation

String-like theory of many-particle Quantum Chaos Boris Gutkin University of Duisburg-Essen Joint work with V. Al. Osipov Luchon, France March 2015 – p. 1

Basic Question Quantum Chaos Theory: Standard semiclassical limit: fixed N (number of particles), � eff → 0 Non-standard: fixed � eff , N → ∞ (Bosons) But what if � eff → 0 and N → ∞ ? – p. 2

Single-particle Quantum Chaos Gutzwiller’s trace formula: � i � � � ρ ( E ) = δ ( E − E n ) ∼ ¯ ρ ( E ) + ℜ A γ exp � S γ ( E ) ���� n γ ∈ PO Smooth � �� � Oscillating A γ stability factor, S γ ( E ) action of a periodic orbit γ γ Number of periodic orbits grows exponentially with length – No prediction on E n from an individual γ – All { γ } together ⇐ ⇒ spectrum – p. 3

Two-point correlation function R ( ε ) = 1 ρ 2 � ρ ( E + ε/ ¯ ρ ) ρ ( E ) � E − 1 ¯ � + ∞ R ( ε ) e − 2 πiτε dε ≈ (Semiclassically) K ( τ ) = −∞ �� �� � ≈ 1 τ − ( T γ + T γ ′ ) i A γ A ∗ � ( S γ − S γ ′ ) δ γ ′ e , T 2 2 T H H γ,γ ′ E T γ , T γ ′ are periods of γ, γ ′ , T H = 2 π � ¯ ρ (Heisenberg time) Spectral correlations ⇐ ⇒ Correlations between actions of periodic orbits – p. 4

Semiclassical origins of universality K ( τ ) = c 1 τ + c 2 τ 2 . . . Diagonal approximation γ = γ ′ = ⇒ Leading order: c 1 [M. Berry 1985] Diagonal approximation Sieber−Richter pairs Sieber-Richter pairs (Non-trivial correlations) = ⇒ Second order: c 2 [M. Sieber K. Richter 2001] ⇒ Duration of encounter ∼ τ E = λ − 1 | log � | S γ − S γ ′ ∼ � = � �� � Ehrenfest time – p. 5

Full theory – all orders in τ S. Müller, S. Heusler, P . Braun, F . Haake, A. Altland 2004 } τ { Σ τ n K( )= Structures of Periodic Orbits Pairing is robust under perturbation Other correlations are washed out! Pairing mechanism = ⇒ Universality of spectral correlations – p. 6

Many-particle systems N p 2 � n H = 2 m + V ( x n ) + V int ( x n − x n +1 ) n =1 Chaos; Local, Homogeneous interactions i.e, invariance under n → n + 1 Many−particle Periodic Orbit Single−particle Periodic Orbit d−dimensions Nd−dimensions 1 2 N Q: Is the single-particle theory of Quantum Chaos applicable? A: Depends how large N is – p. 7

Caricature: Semiclassical Field Theory Continues limit n → η ∈ [0 , ℓ ] , x n ( t ) → x ( η, t ) N x 2 ˙ � n L = 2 m − V ( x n ) − V int ( x n − x n +1 ) − → n =1 � ℓ dη ( ∂ t x ( η, t )) 2 + ( ∂ η x ( η, t )) 2 − V ( x ( η, t )) L = 0 1) PO -are 2D toric surfaces in d -dim space (Rather than 1D lines in N · d -dim) 2) Encounters are “rings” (Rather than 1D stretches) of “width” ∼ λ − 1 | log � eff | – p. 8

Caricature: Semiclassical Field Theory Single-particle structure diagrams: = } { = e e 2 ���� ���� ���� ���� ���� ���� ���� ���� ���� ���� ���� ���� ���� ���� ���� ���� ���� ���� ���� ���� ���� ���� 1 ���� ���� ���� ���� ���� ���� ���� ���� ���� ���� ���� ���� ���� ���� ���� ���� ���� ���� ���� ���� ���� ���� ���� ���� ���� ���� Distinguished by order of encounters Many-particle structure diagrams: Distinguished by order and winding numbers ω of encounters! – p. 9

Many-particle Quantum Chaos λ − 1 | log � eff | =: n E - Ehrenfest “number” T N λ −1 log h < T,N T ������� ������� ������� ������� ������� ������� ������� ������� = ������� ������� ������� ������� log h ������� ������� T< n E N λ ������ ������ ����� ����� ������ ������ ������ ������ ����� ����� ������ ������ ����� ����� ����� ����� ������ ������ ������ ������ ����� ����� ������ ������ ����� ����� ����� ����� ������ ������ ����� ����� −1 −1 n = λ λ [A] [B] [C] log h log h E A. If N � n E , T � n E = ⇒ only ω = (0 , 1) “single-particle” encounters Effectively single-particle Quantum Chaos B. N, T � n E , = ⇒ ω = (0 , 0) encounters dominate! C. N � n E , T � n E = ⇒ only ω = (1 , 0) encounters B, C - Genuine many-particle Quantum Chaos! – p. 10

Coupled-Cat Maps q i+2 q i+1 q i S ( q t , q t +1 ) = S 0 ( q t , q t +1 ) + S int ( q t ) , q t = ( q 1 ,t , q 2 ,t . . . q N ,t ) N uncoupled cat maps, q n,t , p n,t ∈ [0 , 1] : N � S ( n ) S 0 = 0 ( q n,t , q n,t +1 ) + V ( q n,t ) n =1 Interactions: N � S int = − q n,t q 1+( n mod N ) ,t . n =1 – p. 11

Coupled-Cat Maps V = 0 Z t = ( q 1 ,t , p 1 ,t , . . . q N ,t , p N ,t ) ⊺ , Z t +1 = B N Z t mod1 , with 2 N × 2 N matrix B N given by:   A B . . . B 0 0   B A B . . . 0 0          0 B A . . . 0 0  1  1 0 a    , B = − B N = , A =   . . . . . ...     . . . . . ab − 1 0 b b . . . . .       . . . A B 0 0 0     B . . . B A 0 0 Lyapunov exponents: cosh λ k = ( a + b ) / 2 − cos(2 πk/N ) , k = 1 , . . . N Full chaos: | Re λ k | > 0 , i.e. | a + b | > 4 – p. 12

Particle-time Duality Newtonian form: ∆ 2 t q n,t + ∆ 2 n q n,t = ( a + b − 4) q n,t + V ′ ( q n,t ) mod 1 ∆ 2 α - discrete Laplacian: ∆ 2 α f α ≡ f α +1 − 2 f α + f α − 1 Particle-time duality: If { q n,t } solution then { q ′ n,t = q t,n } also solution! = ⇒ ⇒ T -part. PO Γ ′ of period N N -part. PO Γ of period T ⇐ 1) S (Γ) = S (Γ ′ ) 2) [ # of N -particle PO of period T ] = [ # of T -particle PO of ⇒ | det( I − B T N ) | = | det( I − B N period N ] ⇐ T ) | � πmN N − 1 T − 1 � πkT � � � � T 2 4 sin 2 = N 2 4 sin 2 Corollary: N T m =1 k =1 – p. 13

2D Symbolic Dynamics Z t +1 + M t = B N Z t , M t = (m 1 ,t , . . . m N ,t ) ⊺ , m n,t = ( m q n,t , m p n,t ) T 4 4 2 1 2 1 2 3 4 3 1 3 4 2 3 2   2 1 2 1 3 m 1 , 1 m 2 , 1 . . . m N, 1 3 3 4 2 1 2 1 4 2 3 3 4 3 2 1 1 2 1 4 3 4 1 1 1 3 1 1 3   4 2 2 3 1 2 4 1 2 2 1 4 4 4 3 m 1 , 2 m 2 , 2 . . . m N, 2   4 4 4 1 4 3 4 2 1 3 2 4 1 4 1 2 M Γ = 1   3 4 3 2 3 2 4 2 4 . . . 1 3 4 1 4 2 ... . . .   1 3 4 4 4 1 3 1 4 3 3 2 1 2 2 . . . 3 1 4 4 3 4 2 4   4 2 3 4 4 2 3 1 2 1 4 1 4 1 4 3 1 3 2 3 2 2 3 1 1 m 1 ,T m 2 ,T . . . m N,T 1 3 4 4 2 1 4 2 3 1 4 2 3 2 2 3 3 2 1 4 1 4 2 1 2 3 3 4 1 3 3 1 N – p. 14

2D Symbolic Dynamics Z t +1 + M t = B N Z t , M t = (m 1 ,t , . . . m N ,t ) ⊺ , m n,t = ( m q n,t , m p n,t ) T 4 4 2 1 2 1 2 3 4 3   1 3 4 2 3 2 m 1 , 1 m 2 , 1 . . . m N, 1 2 1 2 1 3 3 3 4 2 1 2 1 4 2 3 3 4 3 2 1 1 2 1 4 3 4 1 1 1 3 1 1 3   4 2 2 3 1 2 4 1 2 2 1 4 4 4 3 m 1 , 2 m 2 , 2 . . . m N, 2   4 4 4 1 4 3 4 2 1 3 2 4 1 4 1 2 M Γ = 1   3 4 3 2 3 2 4 2 4 . . . 1 3 4 1 4 2 ... . . .   1 3 4 4 4 1 3 1 4 3 3 2 1 2 2 . . . 3   1 4 4 3 4 2 4 4 2 3 4 4 2 3 1 2 1 4 1 4 1 4 3 1 3 2 3 2 2 3 1 1 m 1 ,T m 2 ,T . . . m N,T 1 3 4 4 2 1 4 2 3 1 4 2 3 2 2 3 3 2 1 4 1 4 2 1 2 3 3 4 1 3 3 1 N 1) Small alphabet (does not grow with N ) 2) Uniqueness: Each PO Γ is uniquely encoded by M Γ Γ can be easily restored from M Γ 3) Locality: r × r square of symbols around ( n, t ) defines position of the n ’th particle at the time t up to error ∼ Λ − r – p. 15

Partner Orbits T T T T A A E E 2 2 D A C D A C B B B A D A E 1 E 1 C E 1 E E E 1 E E 1 E E 2 2 1 2 2 C E 2 E 2 B D E E 1 1 A A N N N N T T A B E E C C B A E E ¯ Γ Γ N N Γ is obtained by reshuffling M Γ M ¯ Note: One encounter is enough, even if time reversal symmetry is broken – p. 16

Example of Partner Orbits T = 50 , N = 70 , a = 3 , b = 2 – p. 17

Example of Partner Orbits 1.0 0.34 0.8 0.32 0.6 p 0.30 p 0.4 0.28 0.2 0.26 0.0 0.06 0.08 0.10 0.12 0.14 0.0 0.2 0.4 0.6 0.8 1.0 q q All the points of Γ = { ( q n,t , p n,t ) } and ¯ Γ = { (¯ q n,t , ¯ p n,t ) } are paired – p. 18