Combinatorial theory of the semiclassical evaluation of transport moments II: Algorithmic approach for moment generating functions

Abstract

Articles you may be interested inCombinatorial theory of the semiclassical evaluation of transport moments. I. Equivalence with the random matrix approach J. Math. Phys. 54, 112103 (2013) Electronic transport through chaotic quantum dots exhibits universal behaviour which can be understood through the semiclassical approximation. Within the approximation, calculation of transport moments reduces to codifying classical correlations between scattering trajectories. These can be represented as ribbon graphs and we develop an algorithmic combinatorial method to generate all such graphs with a given genus. This provides an expansion of the linear transport moments for systems both with and without time reversal symmetry. The computational implementation is then able to progress several orders further than previous semiclassical formulae as well as those derived from an asymptotic expansion of random matrix results. The patterns observed also suggest a general form for the higher orders. C 2013 AIP Publishing LLC.[http://dx

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

“…For higher moments this procedure becomes more involved, though it does lead to interesting combinatorial problems [54,55]. Alternatively, transport diagrams can be generated without recourse to periodic orbits leading to a perturbative expansion of moment generating functions for several orders in the parameter 1/N [4,7,9]. Here instead we develop a combinatorial approach to directly describe all transport diagrams.…”

Semiclassical matrix model for quantum chaotic transport with time-reversal symmetry

“…Consider now the average value of (1) over a certain energy window, P ( i, j) E , this window being small in the classical scale but large in the quantum scale; as → 0, constructive interference is required and the result is determined by correlations: partner trajectories must have almost the same collective action as direct ones. The theory of correlated chaotic trajectories has been discussed in detail in a number of papers [28,29,30,32,36]. Trajectories from correlated sets may differ only in small regions (called encounters) in which the direct ones run nearly parallel or anti-parallel, while the partner ones have crossings.…”

“…A different diagrammatic representation, more convenient, of correlated trajectories uses ribbon graphs [36]. In this case we turn every encounter into a vertex, and trajectories are depicted as edges of ribbons.…”

“…This started to be done perturbatively in [26,27] and was shown to give the exact result for the simplest observables in [28,29,30]. Attention then turned to more general transport statistics [31,32,33,34], until complete equivalence was shown between semiclassics and RMT [35,36] (in the meantime, semiclassics was able to go beyond RMT, incorporating effects due to finite Ehrenfest time, see e.g. [37,38,39]).…”

Energy-dependent correlations in the S-matrix of chaotic systems

“…These restrictions are satisfied in the presence of crossings, analogously to what happens in closed systems. [57,58] The theory is perturbatively formulated in terms of diagrams, as a power series in 1/M , with the contribution of a diagram being proportional to M V −E , where V and E are the numbers of vertices and edges of the diagram, respectively, and with a coefficient which is a rational function of ǫ (although most advances have happened for in the far simpler case of ǫ = 0 [59,60,61]). …”