Semiclassical Foundation of Universalit y in Quantum Chaos

1. Semiclassical Foundation of Universality in Quantum Chaos Sebastian Müller, Stefan Heusler, Petr Braun, Fritz Haake, Alexander Altland preprint: nlin.CD/0401021

2. BGS conjecture Fully chaotic systems have universal spectral statistics on the scale of the mean level spacing Bohigas, Giannoni, Schmit 84

3. Spectral form factor 2 v Ý Þ Ý Þ ? E E _ _ _ X v Ý Þ ? i E E T / Ý Þ v ¤ ? correlations of level density d E E e 2 _ > Ý Þ Ý Þ E E ? E _ = N i i I T 2 = ^ ¤ _ = # H ? f 1 Ý 2 Þ Ý Þ ^ ¤ K b = T 1 b = < T average over and time H v E E + 2 Heisenberg time described by

4. average over ensembles of Hamiltonians yields no TR invariance (unitary class) b K ( ) b = with TR invariance (orthogonal class) Ý Þ ? 2 ln 1 2 b b + b 2 3 ? ? 2 2 2 = b b + b u Random-matrix theory (for t < 1) Why respected by individual systems? Series expansion derived using periodic orbits

5. Gutzwiller trace formula spectral form factor quantum spectral correlations classical action correlations Argaman et al. 93 Periodic orbits Need pairs of orbits with similar action

6. Diagonal approximation 1 without TR invariance U = with TR invariance 2 Berry, 85 orbit pairs: g= g‘ g= time-reversed g‘ (if TR invariant)

7. -2t2in the orthogonal case Sieber/Richter 01, Sieber 02 valid for general hyperbolic systems S.M. 03, Spehner 03, Turek/Richter 03 f>2 in preparation Sieber/Richter pairs

8. l-encounters 1 const. duration tenc i ln ¤ V l orbit stretches close up to time reversal

9. Partner orbit(s) • reconnection inside encounter

10. Partner orbit(s) pose • partner may not decom • reconnection inside encounter

11. Classify & count orbit pairs • number vlofl-encounters > V v # encounters = = l l ³ 2 > # encounter stretches = L l v = l l ³ 2 3 3 Ý Þ N v v number of structures • structure of encounters - stretches time-reversed or not - ordering of encounters - how to reconnect?

12. Classify & count orbit pairs • phase-space differences between encounter stretches probability density Ý Þ w s , u T orbit period phase-space differences

13. u s Phase-space differences Poincaré section piercings • have stable and unstable coordinates s, u • determine: encounter duration, partner, • action difference

14. Phase-space differences use ergodicity: uniform return probability

15. Orbit must leave one encounter ... before entering the next Phase-space differences Overlapping encounters treated as one ... before reentering

16. Orbit must leave one encounter ... before entering the next no reconnection possible Phase-space differences Overlapping encounters treated as one ... before reentering otherwise: self retracing reflection

17. probability density ? L 1 > Ý Þ ? T T lt enc Ý Þ J w s , u T ? L V < t I enc - ergodic return probability 1 I Phase-space differences follows from - integration over L times of piercing - ban of encounter overlap

18. Spectral form factor Berry With HOdA sum rule sum over partners g’ X > ? ? i L V L V S / Ý Þ Ý Þ Ý Þ A ¤ K N v d u d s w u , s e b = U b + U b T v with

19. exit ports entrance ports 1 1 2 2 3 3 Structures of encounters

20. reconnection inside encounters ..... permutation PE 3 Ý Þ N v ..... l-cycle of PE l-encounter ..... permutation PL loops partner must be connected numbers ..... PLPE has only one c cycle ..... structural constants ccccc of perm. group Structures of encounters related to permutation group

21. 3 Ý Þ N v Recursion for Taylor coefficients Ý Þ ? n 1 K 0 unitary = n Ý Þ Ý Þ ? ? ? n 1 K 2 n 2 K orthogonal = n ? n 1 gives RMT result Structures of encounters Recursion fornumbers

22. 3 Ý Þ N v recursion for….. Wick contractions Analogy to sigma-model orbit pairs….. Feynman diagram self-encounter ….. vertex l-encounter….. 2l-vertex external loops ….. propagator lines

23. Conclusions Universal form factor recovered with periodic orbits in all orders Conditions: hyperbolicity, ergodicity, no additional degeneracies in PO spectrum Contribution due to ban of encounter overlap Relation to sigma-model

24. Example: t3-families Need L-V+1 = 3 one 3-encounter two 2-encounters

25. Overlap of two antiparallel 2-encounters

26. Self-overlap of antiparallel 2-encounter

27. Self-overlap of parallel 3-encounter