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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
correlations of level density
_E= > i NE ?E i
described by
v
= Xd
K
b
E ?E 
e
b= TTH < 1
I
T H =2^
_
=
# 2^f?1
Heisenberg time
2
v




?
E
E
_ _
_
v
i E ?
E 
T /
2
_
v
E and time
average over E +
2
Random-matrix theory
average over ensembles of Hamiltonians
yields
b
K ( b) =
no TR invariance
(unitary class)
2 b? bln1 +2 b with TR invariance
= 2b?2b2 +2b3 ?u
(orthogonal class)
(for t < 1)
Why respected by individual systems?
Series expansion derived using periodic orbits
Periodic orbits
Gutzwiller trace formula
_E _+Re >A e iS /
spectral form factor
K
b=
1
TH
>vA A ve

i
S ?S v
/
N T?
T +
T v
2
Need pairs of orbits with similar action
quantum spectral
correlations
classical action
correlations
Argaman et al. 93
Diagonal approximation
Berry, 85
orbit pairs: g= g‘
g= time-reversed g‘ (if TR invariant)
K diag 
b
= UT1
H
U=
>|A | NT ?T 
2
1
without TR invariance
2
with TR invariance
sum rule
=
Ub
Sieber/Richter pairs
-2t2 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-encounters
l orbit stretches close up to time reversal

t
e
duration tenc  1 ln
const.

Partner orbit(s)
 reconnection inside encounter
Partner orbit(s)
 reconnection inside encounter
 partner may not decompose
Classify & count orbit pairs
 number vl of l-encounters
3
v
V => l2 vl =# encounters
L => l vl =# encounter stretches
l2
 structure of encounters
- stretches time-reversed or not
- ordering of encounters
- how to reconnect?
3
number of structures N
v
Classify & count orbit pairs
 phase-space differences
between encounter stretches
probability density
w T s, u 
orbit period
phase-space differences
Poincaré
section
....
Phase-space differences
u
s
piercings
• have stable and unstable coordinates s, u
• determine: encounter duration, partner,
action difference
Phase-space differences
use ergodicity:
uniform return
probability
dt du ds
I
Phase-space differences
Orbit must leave one encounter
... before entering the next
Overlapping encounters treated as one
... before reentering
Phase-space differences
Orbit must leave one encounter
... before entering the next
Overlapping encounters treated as one
... before reentering
otherwise: self retracing reflection
no reconnection possible
Phase-space differences
probability density
wT
s, uJ
T
T? > lt enc L?1
IL?V < t enc
follows from
- ergodic return probability
1
I
- integration over L times of piercing
- ban of encounter overlap
Spectral form factor
Berry
>  With
HOdA sum rule
sum over partners g’
K
b=Ub+Ub> v Nv  Xd L?Vu d L?Vs w T u, se iAS/
3
=k
v
bL?V
3
with k
v =
?1 V

< lVl
L?V?1 
!L

Structures of encounters
entrance ports
1
exit ports
1
2
2
3
3
Structures of encounters
related to permutation group
reconnection inside
encounters
.....
permutation PE
l-encounter
.....
l-cycle of PE
loops
.....
permutation PL
partner must be
connected
v
numbers N 3
..... PLPE has only one
c
cycle
..... structural constants
ccccc of perm. group
Structures of encounters
3
Recursion for numbers N
v
Recursion for Taylor coefficients

n ?1K n = 0
unitary

n ?1K n = ?2n ?2
K n?1 orthogonal
gives RMT result
Analogy to sigma-model
orbit pairs
…..
Feynman diagram
self-encounter
…..
vertex
l-encounter
…..
2l-vertex
external loops
….. propagator lines
recursion for N3
v ….. Wick contractions
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
two 2-encounters
one 3-encounter
Overlap of two antiparallel 2-encounters
Self-overlap of antiparallel 2-encounter
<
<
Self-overlap of parallel 3-encounter
=