Dynamical properties of the 1D Hubbard model using the cumulant Green's functions method

The objective of this work is the theoretical study of the one-band Hubbard model and the introduction of an alternative methodology to solve this model that might be competitive with the current methods used, such as the dynamical mean-field theory [DMFT]. To this purpose, we exactly diagonalize an atomic Hubbard cluster composed of N sites up to N=9 (our computational limit). From the solution of this atomic Hamiltonian, the atomic Green's functions are calculated employing the Lehmann representation and used to obtain the cumulants. Finally, those cumulants are used as "seeds" to find Green's functions for the lattice.
We calculate the relative weight of the residues of the Green’s functions as the transition energy changes, the density of states [DOS] for the lattice, the occupation numbers, and discuss the convergence of the method as the size of the chain increases, comparing it to the Bethe Ansatz (BA) in the particle-hole symmetric case. Results improve with the length of the atomic "seed". For N=9, these results practically coincide with the BA. 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.