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 ∑ ∗ e i S a − S b / F a F b a , b contributions from b = a (and b =T a for TRI) * * a,b identical up to reconnections in encounters gives non-oscillatory part in full but misses oscillatory part
reminder: RMT − 1 ei 2 unitary R 2 2 4 … e i 2 orthogonal 1 − 1 … E − E ′ 2
generating function 〈 det E C − H det E − D − H det E A − H det E − B − H Z A , B , C , D exp tr ln det ∂ A ∂ B Z | A B C D tr ∂ 2 C 1 1 E − H tr E − − H
semiclassical evaluation E det E − H 1 exp dE ′ tr 1 E ′ − H i S a E / − i N E − ∑ a F a e exp exp − i N E ∑ A F A − 1 n A e i S A E / det E − − H det E − H ∗
Riemann-Siegel lookalike allows for real E , enforcing convergence and reality T A T H /2 det E − H exp − i N E ∑ F A − 1 n A e i S A E / c.c. A rigorous for finite matrices, respects unitarity, modelled after Riemann’s ζ not available for inverse determinants
〈 exp i N E A ∑ Z F A e i S A E A / A exp − i N E − B ∑ ∗ e − i S B E − B / F B B T C T H /2 exp − i N E C ∑ F C − 1 n C e i S C E C / C T D T H /2 exp i N E − D ∑ ∗ − 1 n D e − i S D E − D / F D D Z 1 Z 2 + c.c. Weyl Z 2 A , B , C , D Z 1 A , B , − D , − C symmetry
N E N E , S E S E Z 1 e i A B − C − D /2 T C , T D T H /2 ∑ ∗ − 1 n C n D ∗ F C F D 〈 F A F B A , B , C , D e i S A E − S B E S C E − S D E e i A A B B − C C − D D /2 contributions only from terms where orbits in A and C are repeated in either B or D, identically (diagonal appr) or at least up to reconnections in encounters (bunches)
diagonal approximation p.o.’s enter as if uncorrelated 〈 i S a E / average ``sees’’ p.o. sum ∑ a F a 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
1 X e i A B − C − D e Z diag diag X ∑ a F a e i S a E / i A a ∑ b F b ∗ e − i S b E / i B b − ∑ c F c e i S c E / i C c − ∑ d F d ∗ e i S d E / i D d exp 〈 X 2 diag X e diag i A B a − e i A D a ∑ a | F a | 2 e 〈 X 2 diag i C B c − e i C D c − ∑ c | F c | 2 e
∑ a | F a | 2 e i a − ln i const HOdA: e i A B − C − D A D C B 1 Z diag A B C D e i A B C D A − C − D B 2 Z diag A B − D − C e 2 i C diag 2 i 2 − 1 2 i 2
off-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 under weak resolution of configuration space, bunch looks like single orbit orbits in bunch practically identical, apart from reconnections within self-encounters 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 l links encounter stretches can connect links in l ! different ways bunch of l ! (pseudo-)orbits, nearly same ``rigid’’ links; action differences arbitrarily small
bunch of 12
bunch of 72
orbit bunches yield Z 1 asymptotic expansion of in 1/ ε Z 2 Z 1 then from with Riemann Siegel Z 1 Z 2 differentiation of gives correlator R(e), in agreement with RMT; in particular, no addition to diagonal approximation for unitary class n -th order term from bunches of R(e) from bunches with n = L - V +2, where V = # encounters (vertices), L = # links
scene thus set for Sebastian Müller concluding remarks: oscillatory 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
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