Universality in chaotic quantum transport : the concordance between random matrix and semiclassical theories
Gregory Berkolaiko, Texas A&M University
Electronic transport through chaotic quantum dots exhibits universal, system independent properties which are consistent with random matrix theory. The observable quantities can be expressed, via the semiclassical approximation, as sums over the classical scattering trajectories. Correlations between such trajectories are organized diagrammatically and have been shown to yield universal answers for some observables.
We develop a general combinatorial treatment of the semiclassical diagrams by casting them as unicellular maps (graphs embedded on surfaces) and relating them to factorizations of permutations. The expansion of transport quantities in inverse channel number corresponds to a genus expansion of the combinatorial generating function. Taking previously calculated answers (Heusler et al, 2006) for the contribution of a given diagram, we prove agreement between the semiclassical and random matrix approaches to moments of the transmission amplitudes. The proof covers all orders, all moments (including nonlinear), and systems with or without time reversal symmetry. It explains the mathematics behind the applicability of random matrix theory to chaotic quantum transport. The streamlined calculation could also pave the way for inclusion of non-universal effects.
Based on joint work with Jack Kuipers (Regensburg)