Combinatorial theory of the semiclassical evaluation of transport moments. I. Equivalence with the random matrix approach

To study electronic transport through chaotic quantum dots, there are two main theoretical approaches. One involves substituting the quantum system with a random scattering matrix and performing appropriate ensemble averaging. The other treats the transport in the semiclassical approximation and stu...

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Veröffentlicht in:Journal of mathematical physics 2013-11, Vol.54 (11), p.1
Hauptverfasser: Berkolaiko, G., Kuipers, J.
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description To study electronic transport through chaotic quantum dots, there are two main theoretical approaches. One involves substituting the quantum system with a random scattering matrix and performing appropriate ensemble averaging. The other treats the transport in the semiclassical approximation and studies correlations among sets of classical trajectories. There are established evaluation procedures within the semiclassical evaluation that, for several linear and nonlinear transport moments to which they were applied, have always resulted in the agreement with random matrix predictions. We prove that this agreement is universal: any semiclassical evaluation within the accepted procedures is equivalent to the evaluation within random matrix theory. The equivalence is shown by developing a combinatorial interpretation of the trajectory sets as ribbon graphs (maps) with certain properties and exhibiting systematic cancellations among their contributions. Remaining trajectory sets can be identified with primitive (palindromic) factorisations whose number gives the coefficients in the corresponding expansion of the moments of random matrices. The equivalence is proved for systems with and without time reversal symmetry.
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subjects Approximation
CHAOS THEORY
CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS
Combinatorial analysis
Correlation analysis
DIAGRAMS
Electron transport
Equivalence
EVALUATION
GRAPH THEORY
MATRICES
Matrix
Matrix theory
NONLINEAR PROBLEMS
Physics
QUANTUM DOTS
Quantum theory
RANDOMNESS
SCATTERING
SEMICLASSICAL APPROXIMATION
SYMMETRY
Thermal expansion
TRAJECTORIES
Trajectory analysis
title Combinatorial theory of the semiclassical evaluation of transport moments. I. Equivalence with the random matrix approach
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