Tau-Function Theory of Quantum Chaotic Transport with beta=1,2,4

TL;DR

This paper develops a comprehensive integrable theory for quantum chaotic transport in ballistic quantum dots, deriving differential equations and recurrence relations for cumulants across all symmetry classes, and linking delay time statistics to Painleve' transcendents.

Contribution

It introduces a unified approach to compute cumulants and their generating functions for all symmetry classes beta=1,2,4, including weak localization corrections and connections to Painleve' equations.

Findings

Derived differential equations for cumulant generating functions.

Proved conjectures on weak localization corrections.

Established recurrence relations for cumulant computation.

Abstract

We study the cumulants and their generating functions of the probability distributions of the conductance, shot noise and Wigner delay time in ballistic quantum dots. Our approach is based on the integrable theory of certain matrix integrals and applies to all the symmetry classes beta=1,2,4 of Random Matrix Theory. We compute the weak localization corrections to the mixed cumulants of the conductance and shot noise for beta=1,4, thus proving a number of conjectures of Khoruzhenko et al. (Phys. Rev. B, Vol. 80 (2009), 125301). We derive differential equations that characterize the cumulant generating functions for all beta=1,2,4. Furthermore, we show that the cumulant generating function of the Wigner delay time can be expressed in terms of the Painleve' III' transcendant. This allows us to study properties of the cumulants of the Wigner delay time in the asymptotic limit n -> infinity. Finally, for all the symmetry classes and for any number of open channels, we derive a set of recurrence relations that are very efficient for computing cumulants at all orders.