The cumulant Green's functions method for the single impurity Anderson model

TL;DR

This paper introduces a cumulant Green's functions method (CGFM) for the single impurity Anderson model, providing an exact solution approach that accurately captures low-temperature electronic properties without self-consistency, and benchmarks well against NRG results.

Contribution

The paper presents a novel CGFM approach that simplifies solving SIAM by avoiding self-consistency, accurately capturing low-temperature physics, and demonstrating potential for broader strongly correlated systems.

Findings

Accurate density of states and impurity occupation numbers at low temperatures.

Only four atomic transitions dominate the SIAM density of states across regimes.

CGFM matches NRG results and offers a promising alternative for strongly correlated models.

Abstract

Using the cumulant Green's functions method (CGFM), we study the single impurity Anderson model (SIAM). The CGFM starting point is a diagonalization of the SIAM Hamiltonian expressed in a semi-chain form, containing N sites, viz., a correlated site (simulating an impurity) connected to the remaining N-1 uncorrelated conduction-electron sites. An exact solution can be obtained since the complete system has few sites. That solution is employed to calculate the atomic Green's functions and the approximate cumulants used to obtain the impurity and conduction Green's functions for the SIAM, and no self-consistency loop is required. We calculated the density of states, the Friedel sum rule, and the impurity occupation number, all benchmarked against results from the numerical renormalization group (NRG). One of the main insights obtained is that, at very low temperatures, only four atomic transitions contribute to generating the entire SIAM density of states, regardless of the number of sites in the chain and the model's parameters and different regimes: Empty orbital, mixed-valence, and Kondo. We also pointed out the possibilities of the CGFM as a valid alternative to describe strongly correlated electron systems like the Hubbard and t-J models, the periodic Anderson model, the Kondo and Coqblin-Schrieffer models, and their variants.