Quantum Chaos : Learning from network models Quantum graphs, weak localisation and RMT archive : nlin.CD/0107056

Gregory Berkolaiko1 2 , Holger Schanz3 & Rob Whitney1 4 1 Weizmann Institute of Science, Israel 2 University of Strathclyde 3 Universitat ¨ Gottingen, ¨ Germany 4 University of Oxford

[1] Quantum Chaos and RMT Bohigas-Gianonni-Schmit (BGS) Conjecture ’84 : Quantum systems with chaotic classical dynamics have random matrices level-statistics. time-reversal symmetry

)

GOE level-statistics

Integrable

1111111 0000000 0000000 1111111 1111111 0000000 1111111 0000000

Chaotic

1111111 0000000 0000000 1111111 1111111 0000000 1111111 0000000

Form factor

K ( ) = FT of level-level correlation funct.

KRMT ( ) = 2 =2

; log(1 + 2 ) ; 2 + 2 + O 2

3

4

]

K

Objective : Find ( ) for chaotic system thereby prove or disprove BGS-conjecture Hard problem — forced to work with toy models

[2] Periodic orbit theory Gutzwiller ’71 : ray-optics limit of quantum mechanics

h

Feynman path integral for small  Sum over classical orbits P & Q

;!

* X 1

K ( ) = AP AQ e hi (SP ;SQ ) PQ

where

+

hi indicates Energy Averaging is mean level spacing and = t =h 

Berry diagonal approximation ’85 :

Q=P

strictly it is any cyclic permutation of P

K ( ) = 2 + 

 

No systematic correction scheme Corrections within periodic orbit theory : YES Corrections beyond periodic orbit theory : ?

[3] Diag. approx and weak localisation Semiclassical limit (

S
 h ) 1

ri

rf

2

 iS =h  2 iS2 =h  1 P (ri ! rf ) = A1 e + A2 e 

= A21 + A22 + 2A1 A2 cos (S1 ; S2 )=h]

S

S

Diag. Approx : IGNORE correlations between 1 and 2 then with averaging :

P (ri ! rf ) ' A21 + A22

Correlations may be rare 1 but they exist

ri

!

2 rf

Paths of this type weak localisation in disordered systems

[4] Quantum Graphs Kottos-Smilansky ’99 Periodic orbit theory is exact Numerics show GOE level-statistics

B Bonds N Nodes Elements of quantum graph (1)

BONDS : 1-D wave guide

i (x t) = eip(x;x0 )+iEt i (x0  0)

Lengths of all bonds are incommensurate (2)

1

2 3 4

Form factor

NODES : Unitary Scatterers

i out = ij j in P Unitarity : j ij jk = ik

) double sum over periodic orbits X K ( ) = B1 AP AQ LP LQ PQ

(sum over all cyclic permutations of each orbit)

[5] Diagonal approximation and beyond Diagonal approximation for graphs :

X PQ

AP AQ LP LQ ;! t

Q=P

X P

jAP j

2

...and BEYOND Corrections to diag. approx. — weak localisation

P and Q identical everywhere except at intersections. Do systematic expansion in number of intersections.

Topologies of a pair of paths with single intersection

(i)

(ii)

c

a α b

a

c

a

α

α

b

d

b

d

a

c

a

c

α

d

c

b

α d

b

d

[6] Naive estimate Estimate of leading correction to diag. approx. for the following simple graph

B = 21 N 2 (
) with “s-wave” nodes: jab j = 1=N

maximally-connected graph

Classical weight of single-intersection compared to zero-intersections

= t2  N1

# ways to place intersection One node at intersection constrained

Naive estimate of their contribution to form factor

 1 t (1) Kestimate ( )  B  t2  N = N 3

This naive estimate is very WRONG !!

3

 Not a -correction  For  1 and N ! 1 then Kestimate( ) ! 1 2

(1)

[7] Basics of calculation Period orbit t’-2

a

c

P

Period orbit

t’-2

α b

d

t-t’-2

a

c

Q

α b

d

t-t’-2

t;3 X 2 X t K1 ( ) = B M
t ;1a b
M
t;tc;d1
t =3
a b c d  da(
) bc(
) ca(
) db(
) 0

0

0

Assume Graph is ergodic

1 t ) tlim M = !1 m l ml B 0

0

 t ! 1 means t  classical ergodic time  Either t0 or t ; t0 satisfy this limit. ) above sum can be evaluated N

2 3] ...still get WRONG answer [result unless careful about special cases

[8] Special cases and exceptions (i) Must not double-count orbits of the form:

P =  ! c !  ! d !  !  ! a !  ! b !  !  Q =  ! d !  ! c !  !  ! a !  ! b !  ! 

(ii) Must not count orbits where one loop self-retraces.

(i)

c

c

a

a β

α

α

β

b

b d

d c

(ii) δ

γ

β

α

c ε

d

δ

γ

β

α

d

[9] Order

3

contributions

 Without time-reversal symmetry

 With time-reversal symmetry Above diagrams plus

[10] Summary

 Periodic orbit theory on quantum graphs  Calculate form factor : K ( )

(Fourier trans. of level-level correlation function)

 We go beyond the diagonal approximation

 Expansion in # of intersections of path with itself.

Analogous to weak localisation in disordered syst.

 We show mixing (ergodic) graph has  2 ; 2 +  K( ) = 2

+ 0 + 0 + 

diag. approx. beyond diag. approx. “systematic” correction scheme

 Agreement with RMT

beyond the diagonal approximation