Scattering in quantum dots via noncommutative rational functions

TL;DR

This paper develops a new theoretical framework for analyzing the transmission eigenvalue density in quantum dots, extending previous models to more general conditions and revealing novel singularity behaviors.

Contribution

It introduces a general approach using noncommutative rational functions to study spectral densities in large random matrices, broadening the scope of quantum dot transport models.

Findings

Recovers known density formulas in specific limits.

Shows the persistence of singularities for certain ratios.

Identifies a new anomalous singularity at the critical ratio.

Abstract

In the customary random matrix model for transport in quantum dots with $M$ internal degrees of freedom coupled to a chaotic environment via $N\ll M$ channels, the density $\rho$ of transmission eigenvalues is computed from a specific invariant ensemble for which explicit formula for the joint probability density of all eigenvalues is available. We revisit this problem in the large $N$ regime allowing for (i) arbitrary ratio $\phi := N/M \le 1$; and (ii) general distributions for the matrix elements of the Hamiltonian of the quantum dot. In the limit $\phi \to 0$ we recover the formula for the density $\rho$ that Beenakker (Rev. Mod. Phys., 69:731-808, 1997) has derived for a special matrix ensemble. We also prove that the inverse square root singularity of the density at zero and full transmission in Beenakker's formula persists for any $\phi <1$ but in the borderline case $\phi=1$ an anomalous $\lambda^{-2/3}$ singularity arises at zero. To access this level of generality we develop the theory of global and local laws on the spectral density of a large class of noncommutative rational expressions in large random matrices with i.i.d. entries.