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

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

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

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

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

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

Sieber/Richter pairs -2    in 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

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

 reconnection inside encounter Partner orbit(s)

 reconnection inside encounter pose  partner may not decom Partner orbit(s)

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

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

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

Phase-space differences use ergodicity: uniform return probability

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

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

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

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

Structures of encounters entrance ports exit ports 1 2 3

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

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

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

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

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

Overlap of two antiparallel 2-encounters

Self-overlap of antiparallel 2-encounter

Self-overlap of parallel 3-encounter