String-like theory of many-particle Quantum Chaos

Transcription

1 String-like theory of many-particle Quantum Chaos Boris Gutkin University of Duisburg-ssen Joint work with V. Al. Osipov Luchon, France March 05 p.

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

3 Single-particle Quantum Chaos Gutzwiller s trace formula: ρ() = n δ( n ) ρ() }{{} Smooth +R ( ) i A γ exp S γ() γ PO }{{} Oscillating A γ stability factor, S γ () action of a periodic orbit γ γ umber of periodic orbits grows exponentially with length o prediction on n from an individual γ All {γ} together spectrum p.

4 Two-point correlation function R(ε) = ρ ρ( +ε/ ρ)ρ() K(τ) = T H + R(ε)e πiτε dε (Semiclassically) A γ A γ e i (S γ S ) γ δ γ,γ T γ,t γ are periods of γ,γ, ( τ (T ) γ +T γ ) T H T H = π ρ (Heisenberg time) Spectral correlations Correlations between actions of periodic orbits, p.

5 Semiclassical origins of universality K(τ) = c τ +c τ... Diagonal approximation γ = γ = Leading order: c [M. Berry 985] Diagonal approximation Sieber Richter pairs Sieber-Richter pairs (on-trivial correlations) = Second order: c [M. Sieber K. Richter 00] S γ S γ = Duration of encounter τ = λ log }{{} hrenfest time p. 5

6 Full theory all orders in τ S. Müller, S. Heusler, P. Braun, F. Haake, A. Altland 00 Σ Structures of Periodic Orbits{ K( τ )= }τ n Pairing is robust under perturbation Other correlations are washed out! Pairing mechanism = Universality of spectral correlations p. 6

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

8 Caricature: Semiclassical Field Theory Continues limit n η [0,l], L = L = n= l 0 x n (t) x(η,t) ẋ n m V(x n) V int (x n x n+ ) dη( t x(η,t)) +( η x(η,t)) V(x(η,t)) ) PO -are D toric surfaces in d-dim space (Rather than D lines in d-dim) ) ncounters are rings (Rather than D stretches) of width λ log eff p. 8

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

10 Many-particle Quantum Chaos λ log eff =: n - hrenfest number [A] T λ log h = log h λ T [B] λ log h < T, [C] n = λ log h T<n A. If n, T n = only ω = (0,) single-particle encounters ffectively single-particle Quantum Chaos B.,T n, = ω = (0,0) encounters dominate! C. n, T n = only ω = (,0) encounters B, C - Genuine many-particle Quantum Chaos! p. 0

11 Coupled-Cat Maps q i q i+ q i+ S(q t,q t+ ) = S 0 (q t,q t+ )+S int (q t ), q t = (q,t,q,t...q,t ) uncoupled cat maps, q n,t,p n,t [0,]: Interactions: S 0 = n= S (n) 0 (q n,t,q n,t+ )+V(q n,t ) S int = n= q n,t q +(nmod),t. p.

12 Coupled-Cat Maps V = 0 Z t+ = B Z t mod, Z t = (q,t,p,t,...q,t,p,t ), with matrix B given by: B = A B B B A B B A A B,A = a ab b,b = 0 b 0 B B A Lyapunov exponents: coshλ k = (a+b)/ cos(πk/), k =,... Full chaos: Reλ k > 0, i.e. a+b > p.

13 ewtonian form: Particle-time Duality tq n,t + nq n,t = (a+b )q n,t +V (q n,t ) mod α- discrete Laplacian: αf α f α+ f α +f α Particle-time duality: If {q n,t } solution then {q n,t = q t,n } also solution! = -part. PO Γ of period T T-part. PO Γ of period ) S(Γ) = S(Γ ) ) [# of -particle PO of period T] = [# of T-particle PO of period ] det(i B T ) = det(i B T ) Corollary: T k= sin ( πkt ) T = m= sin ( πm T ) p.

14 D Symbolic Dynamics Z t+ +M t = B Z t,m t = (m,t,...m,t ),m n,t = (m q n,t,m p n,t) T M Γ = m, m,... m, m, m,... m, m,t m,t... m,t p.

15 D Symbolic Dynamics Z t+ +M t = B Z t,m t = (m,t,...m,t ),m n,t = (m q n,t,m p n,t) T M Γ = m, m,... m, m, m,... m, m,t m,t... m,t ) Small alphabet (does not grow with ) ) Uniqueness: ach PO Γ is uniquely encoded by M Γ Γ can be easily restored from M Γ ) 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. 5

16 Partner Orbits T A A D B C T A B C D A A B C D A T A A T D C B Γ C T B A Γ T C A B M Γ is obtained by reshuffling M Γ ote: One encounter is enough, even if time reversal symmetry is broken p. 6

17 xample of Partner Orbits T = 50, = 70, a =, b = p. 7

18 xample of Partner Orbits p p q q All the points of Γ = {(q n,t,p n,t )} and Γ = {( q n,t, p n,t )} are paired p. 8

19 Distances between paired points t n d n,t = (q n,t q n,t ) +(p n,t p n,t ), Lardgest distances 0 are between points in encounters p. 9

20 Action differences t A Γ B Γ p n (q (),p ()) n,t n,t (q (),p ()) n,t n,t e C a) n b) C ncounter regions C B A c) (q (),p ()) n,t n,t (q (),p ()) n,t n,t q n S(Γ) S( Γ) = (n,t) C S n,t + (n,t) C S n,t, S a n,t a {, } - symplectic area of the region formed by the points (q n,t (k),p a n,t(k)), ( q n,t (k), p a n,t(k)), k =, S(Γ) S( Γ) independent of C choice as long as it is inside of the encounter p. 0

21 Quantisation Hannay, Berry (980); Keating (99) U is L L unitary matrix, L = eff Translational symmetries: = subspectra approximately of the same size = L /. Almost all are paired i.e., mostly doubly degenerate levels Gutzwiller trace formula Rivas, Saraceno, A. de Almeida (000) Tr(U ) T = det(b T ) Γ PO exp( iπls Γ ). All entries are symmetric under exchange T p.

22 Quantum Duality Particle-time duality (Quantum): Tr(U ) T = Tr(U T ) Form Factor: K (T) = L For short times T < n = λ logl, Regime dual to universal: Tr(U ) T L T K (T) = L T K β (T/L T ) In particular for very short times L T /T <, K β K (T) L T /L Short time exponential growth instead of linear T/L p.

23 Many-particle Semiclassics ( L T ) K (T) = K diag (,T)+K off (,T). Diagonal: K diag (,T) = /β Off-diagonal: K off = χ χ {All possible structures { For a given encounter type ω: Γ, Γ χ A Γ e is Γ S Γ eff K (ω) off (,T) = k= α (k) ω ( T L d ω ) k The scale L d ω is defined by length of encounter: d (0,) =, d (,0) = T, d (0,0) = n For non-interacting particles d =. p.

24 T Summary n Single particle Quantum Chaos o partners n Dual regime t Terra incognita: Many particle Quantum Chaos n K = T Duality: Tr(U ) T K(,T) = K(T,) Challenge: Contributions from new partners. Applications beyond spectral correlations. xtension to: Hamiltonian flows (continues T) Quantum Field Theory (continues T,) p.

25 Summary Sadly, searching for periodic orbits will never become as popular as a week on Côte d Azur, or publishing yet another log-log plot in Phys. Rev. Letters. P. Cvitanović, et al., Chaos: Classical and Quantum Preprint: arxiv: p. 5