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

Journal Article · Fri Nov 15 04:00:00 EST 2013 · Journal of Mathematical Physics

DOI:https://doi.org/10.1063/1.4826442· OSTI ID:22251491

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.

• OSTI ID: 22251491
• Journal Information: Journal of Mathematical Physics, Journal Name: Journal of Mathematical Physics Journal Issue: 11 Vol. 54; ISSN JMAPAQ; ISSN 0022-2488
• Country of Publication: United States
• Language: English

Related Subjects

71 CLASSICAL AND QUANTUM MECHANICS
GENERAL PHYSICS
CHAOS THEORY
DIAGRAMS
EVALUATION
GRAPH THEORY
MATRICES
NONLINEAR PROBLEMS
QUANTUM DOTS
RANDOMNESS
SCATTERING
SEMICLASSICAL APPROXIMATION
SYMMETRY
TRAJECTORIES