The cumulant Green's functions method for the Hubbard model

In this work, we develop a theoretical study of the one-band Hubbard model. We introduce an alternative methodology to solve this model that might be competitive with the current methods, such as the dynamical mean-field theory [DMFT] or theoretical calculations employing different atomic clusters. We exactly diagonalize a linear atomic Hubbard cluster composed of N sites until N=9 (our computational limit). Utilizing the Lehmann representation, the atomic Green's functions are calculated and used to obtain the atomic cumulants ("seeds"). Finally, these cumulants are used as approximations to find Green's functions for the lattice. 

We calculate the density of states [DOS], the ground-state energy, and the occupation numbers. We compare the gap size results in the DOS and the ground-state energy with the 1D exact solution of the Bethe Ansatz (BA) in the particle-hole symmetric case. The results improve with the length of the atomic "seed." For N=9, these results practically coincide with the BA one. The method should be applied in any parameter regime of the model and 2D and 3D as well. The method should also be extended to other strongly correlated models like the periodic Anderson model.