A perturbation theory for the Anderson model

TL;DR

This paper develops a perturbation theory for the Anderson model using a diagrammatic real-time approach, demonstrating the analyticity of the stationary density matrix and current in the coupling parameter, and providing explicit sixth-order kernel results.

Contribution

It introduces a convergent power series expansion for kernels in the Anderson model and proves the analyticity of key quantities, with explicit sixth-order kernel calculations.

Findings

Kernels form a convergent power series in coupling w

Stationary density matrix and current are analytic in w

Zero bias resonance becomes more pronounced at low temperature

Abstract

Within the diagrammatic real time approach \cite{K\"onig96, Schoeller97}, the current across a quantum dot which is tunnel coupled to two leads at different chemical potentials is calculated by the use of two objects referred to as kernels. The stationary reduced density matrix of the quantum dot is determined by the use of the density matrix kernel, while the current kernel is used in a second step to determine the stationary current across the dot. If the tunneling Hamiltonian is multiplied by a coupling parameter ``w``, then everything, including the kernels, the stationary density matrix as well as the stationary current, can be viewed as a function of w. In the time space, and at every single and fixed time t, the kernels have the clear structure of a convergent power series in w. Refer to the coefficients of these power series as the orders of the kernels. It is intuitive to truncate the kernels at some finite order and to perform remaining calculations by the use of the corresponding approximate kernels. However, the quantities which actually appear in the calculations are not the kernels as a function of time but rather their Laplace transforms, and here only in the limit \lambda \to 0. The statement that even in this limit the structure of a convergent power series in the coupling parameter is still conserved is shown in the text. The statement that the stationary density matrix and current are still analytic in the coupling parameter is shown, assuming the quantum dot is the single impurity Anderson model (SIAM). Finally, results for the kernels up to sixth order, neglecting the doubly occupied state and assuming equal energies E_\uparrow= E_\downarrow, are presented and discussed. In case the degenerate level lies below the Fermi level, a zero bias resonance, getting more and more pronounced with decreasing temperature, is expected.