Numéro |
J. Phys. I France
Volume 5, Numéro 7, July 1995
|
|
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Page(s) | 877 - 906 | |
DOI | https://doi.org/10.1051/jp1:1995111 |
J. Phys. I France 5 (1995) 877-906
Brownian Motion Ensembles and Parametric Correlations of the Transmission Eigenvalues: Application to Coupled Quantum Billiards and to Disordered Wires
Klaus Frahm and Jean-Louis PichardService de Physique de l'État Condensé, CEA Saclay, 91191 Gif-sur-Yvette, France
(Received 20 November 1994, accepted 7 March 1995)
Abstract
The parametric correlations of the transmission eigenvalues
Ti of a
N-channel quantum scatterer
are calculated assuming two different Brownian motion ensembles. The first one is the original
ensemble introduced by Dyson and assumes an isotropic diffusion for the
S-matrix. We derive the
corresponding Fokker-Planck equation for the transmission eigenvalues, which can be mapped for the
unitary case onto an exactly solvable problem of
N non-interacting fermions in one dimension with
imaginary time. We recover for the
Ti the same universal parametric correlation than the ones
recently obtained for the energy levels, within certain limits. As an application, we consider
transmission through two chaotic cavities weakly coupled by a
n-channel point contact when a
magnetic field is applied. The
S-matrix of each chaotic cavity is assumed to belong to the Dyson
circular unitary ensemble (CUE) and one has a
one CUE crossover when
n
increases. We calculate all types of correlation functions for the transmission eigenvalues
Ti
and we get exact finite
N results for the averaged conductance
and its
variance
, as a function of the parameter
n. The second Brownian motion
ensemble assumes for the transfer matrix
M an isotropic diffusion yielded by a multiplicative
combination law. This model is known to describe a disordered wire of length
L and gives another
Fokker-Planck equation which describes the
L-dependence of the
Ti. An exact solution of this
equation in the unitary case has recently been obtained by Beenakker and Rejaei, which gives their
L-dependent joint probability distribution. Using this result, we show how to calculate all types
of correlation functions, for arbitrary
L and
N. This allows us to get an integral expression
for the average conductance which coincides in the limit
with the microscopic non
linear
-model results obtained by Zirnbauer et al., establishing the equivalence of
the two approaches. We review the qualitative differences between transmission through two weakly
coupled quantum dots and through a disordered line and we discuss the mathematical analogies between
the Fokker-Planck equations of the two Brownian motion models.
© Les Editions de Physique 1995