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

Using the cumulant Green's functions method (CGFM), we study the single impurity Anderson model (SIAM). The CGFM starting point is the 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 generate 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.