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Semiclassical Foundation of Universalit y in Quantum Chaos. Sebastian M ü ller, Stefan Heusler, Petr Braun, Fritz Haake, Alexander Altland. preprint: nlin.CD/0401021. BGS conjecture. Fully chaotic systems have universal spectral statistics. on the scale of the mean level spacing.
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Semiclassical Foundation of Universality in Quantum Chaos Sebastian Müller, Stefan Heusler, Petr Braun, Fritz Haake, Alexander Altland preprint: nlin.CD/0401021
BGS conjecture Fully chaotic systems have universal spectral statistics on the scale of the mean level spacing Bohigas, Giannoni, Schmit 84
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
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
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
Diagonal approximation 1 without TR invariance U = with TR invariance 2 Berry, 85 orbit pairs: g= g‘ g= time-reversed g‘ (if TR invariant)
-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
l-encounters 1 const. duration tenc i ln ¤ V l orbit stretches close up to time reversal
Partner orbit(s) • reconnection inside encounter
Partner orbit(s) pose • partner may not decom • reconnection inside encounter
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?
Classify & count orbit pairs • phase-space differences between encounter stretches probability density Ý Þ w s , u T orbit period phase-space differences
u s Phase-space differences Poincaré section piercings • have stable and unstable coordinates s, u • determine: encounter duration, partner, • action difference
Phase-space differences use ergodicity: uniform return probability
Orbit must leave one encounter ... before entering the next Phase-space differences Overlapping encounters treated as one ... before reentering
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
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
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
exit ports entrance ports 1 1 2 2 3 3 Structures of encounters
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
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
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
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
Example: t3-families Need L-V+1 = 3 one 3-encounter two 2-encounters