Semiclassical Foundation of Universalit y in Quantum Chaos
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Transcript of Semiclassical Foundation of Universalit y in Quantum Chaos
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
described by
Spectral form factor
dE Ee i E E T /EE 2
2K
correlations of level density
E i E E i
E E
2average over and time
T 1
Heisenberg time
TH
TH 2 2 f1
Random-matrix theory
Why respected by individual systems?
Series expansion derived using periodic orbits
yields
average over ensembles of Hamiltonians
K ( )
no TR invariance (unitary class)
2 ln1 2 with TR invariance (orthogonal class)
for < 1)
2 22 23
Periodic orbits
Need pairs of orbits with similar action
quantum spectral correlations
classical action correlations
Argaman et al. 93
Gutzwiller trace formula
E Re A e iS /
spectral form factor
K 1TH
A A e iS S / T
T T
2
orbit pairs:‘
Diagonal approximation Berry, 85
1
2
without TR invariance
with TR invariance
Kdiag 1TH
|A |2 T T sum rule
time-reversed‘ (if TR invariant)
Sieber/Richter pairs
-2in the orthogonal caseSieber/Richter 01, Sieber 02
valid for general hyperbolic systemsS.M. 03, Spehner 03, Turek/Richter 03f>2 in preparation
l orbit stretches close up to time reversal
l-encounters
e t duration tenc 1 ln const.
reconnection inside encounter
Partner orbit(s)
reconnection inside encounter pose partner may not decom
Partner orbit(s)
lV 2 vl # encounters
l 2L
l vl # encounter stretches
structure of encounters
- ordering of encounters
number vlofl-encounters v
Classify & count orbit pairs
- stretches time-reversed or not
- how to reconnect?
Nv number of structures
Classify & count orbit pairs
phase-space differencesbetween encounter stretches
probability density
w T s , u
orbit periodphase-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:
dt du ds
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 reenteringotherwise: self retracing reflection
no reconnection possible
Orbit must leave one encounter... before entering the next
Phase-space differences
- ban of encounter overlap
probability density
wTs, u TT ltenc L1
LV tenc
1
- ergodic return probability
follows from
- integration over L times of piercing
BerryWith HOdA sum rule
sum over partners ’
K v Nv dLVu dLVs wTu, s eiS/
Spectral form factor
kv L V
kv 1 V lVl
L V 1 ! Lwith
Structures of encounters
entrance ports
1
2
3
exit ports
1
2
3
Structures of encounters
related to permutation group
reconnection insideencounters
..... permutation PE
l-encounter ..... l-cycle of PE
loops ..... permutation PL
partner must be connected
..... PLPE has only one c cycle
numbers ..... structural constants ccccc of perm. group
Nv
Structures of encounters
n 1K n 0 unitary
n 1K n 2n 2K n 1 orthogonal
Recursion for numbers
Recursion for Taylor coefficients
gives RMT result
Nv
Analogy to sigma-model
orbit pairs ….. Feynman diagram
self-encounter ….. vertex
l-encounter ….. 2l-vertex
external loops ….. propagator lines
recursion for ….. Wick contractions
Nv
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-familiesNeed 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
=