Generating function for level correlations, semiclassical - - PowerPoint PPT Presentation

Generating function for level correlations, semiclassical evaluation

Alex Altland, Peter Braun, F. H., Stefan Heusler, Sebastian Müller

Banff, Feb. 25, 2008

conventional semiclassics

〈EE ′ −  2   ∑

a,b

F aF b

∗e iS a−S b/

contributions from b=a (and b=Ta for TRI)

* * a,b identical up to reconnections in encounters

gives non-oscillatory part in full but misses oscillatory part

reminder: RMT

  E − E ′2 

R  

−1  ei2 2

−1 …  

1 24 …ei2

unitary

• rthogonal

generating function 〈 detEC−HdetE−D−H

detEA−HdetE−B−H   ZA,B,C,D

det  exp tr ln

∂2 ∂A∂B Z|ABCD  tr 1 E−Htr 1 E−−H

 C

semiclassical evaluation

 exp −iNE − ∑ a F ae

iSaE /

 exp−iNE  ∑ A FA −1 nAeiSAE / detE  − H  exp 

E

dE ′ tr

1 E′−H

   1 

 

detE − − H  detE − H∗

detE − H  exp−iNE∑

A

F A −1 nA eiSAE/

allows for real E, enforcing convergence and reality

T A  TH/2

 c.c.

Riemann-Siegel lookalike

rigorous for finite matrices, respects unitarity, modelled after Riemann’s ζ not available for inverse determinants



Z  expiNEA∑

A

FA eiSAEA/ exp−iNE−B∑

B

FB

∗ e−iSBE−B/

 exp−iNEC ∑

C TC TH/2

FC −1nC eiSCEC/  expiNE−D ∑

D TD  TH/2

FD

∗ −1nD e−iSDE−D/

〈



+ c.c.



 Z 1 Z 2 Z 2A,B, C,D  Z1 A,B,−D,− C

Weyl symmetry

contributions only from terms where orbits in A and C are repeated in either B or D, identically (diagonal appr)

• r at least up to reconnections in encounters (bunches)

Z1  eiAB−C−D/2 

∑

A,B,C,D TC,TD TH/2

〈FAFB

∗F CFD ∗ −1nC nD

 eiSAE − SBE  SCE − SDE  eiAA  BB − CC − DD/2

NE    NE  , SE    SE  

diagonal approximation

p.o.’s enter as if uncorrelated average ``sees’’ p.o. sum ∑ a F ae

iSaE /

〈

as Gaussian random variable, due to central limit theorem Gaussian average most conveniently done in starting expression where four p.o. sums appear in exponent

X  ∑ a Fae iSaE/iAa  ∑b Fb

∗e −iSbE/iBb

− ∑ c F ceiScE/iCc − ∑ d Fd

∗eiSdE/iDd

e

X diag

 exp 〈X2 diag

〈X 2 diag 

∑ a|F a|2 e

iABa − e iADa

− ∑ c|Fc| 2 e

iCBc − e iCDc

Z diag

1

 eiAB−C−D e

X diag

HOdA:

∑ a|Fa| 2eia  − lni   const

Z diag

1

 eiAB−C−D AD CB

ABCD 

Z diag

2

 eiABCD A−C −DB

AB−D−C 

C diag 

1 2i2 − e2i 2i2

• ff-diagonal terms from orbit bunches

in chaotic dynamics, long p. o.’s do not arise as mutually independent entities but rather in closely packed bunches

• rbits in bunch practically identical, apart

from reconnections within self-encounters under weak resolution of configuration space, bunch looks like single orbit all orbits in bunch generated from single one by reshuffling stretches within self-encounters, so as to differently connect practically unchanged links

``bunch’’ of 2 orbits: nearly same links, differently connected by the two encounter stretches; action difference can be arbitrarily small

respectful bows to Martin & Klaus, and their “disordered precursors”

l-encounter: l orbit stretches mutually close

encounter stretches can connect links in l! different ways bunch of l! (pseudo-)orbits, nearly same ``rigid’’ links; action differences arbitrarily small l links

bunch of 12

bunch of 72

• rbit bunches yield

asymptotic expansion of in 1/ε

Z 1

n-th order term from bunches of R(e) from bunches with n = L-V+2, where then from with Riemann Siegel

Z1

Z 2 differentiation of gives correlator R(e),

Z 1  Z2

in agreement with RMT; in particular, no addition to diagonal approximation for unitary class V = # encounters (vertices), L = # links

scene thus set for Sebastian Müller concluding remarks:

• scillatory terms through generating function

and Riemann-Siegel close correspondence to sigma model of RMT asymptotic series for osc and non-osc terms there arise from perturbative treatment of two saddle points for integral over matrix manifold, Feynman diagrams correspond to orbit bunches, vertices to encounters, links to propagator lines