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

=