Self-consistent solution of the Hubbard model on a 4x4 cluster with the parquet formalism

A self-consistent solution of the Hubbard model is performed on a 4x4 cluster at both the one and the two-particle level. We combine the Parquet and the Bethe-Salpeter equations into one non-linear equation to take advantage of optimized linear solvers such as GMRES and BICG-Stab. We calculate some relevant quantities and compare them to the results obtained from Determinant Quantum Monte Carlo (DQMC), self-consistent second order approximation and FLuctuation EXchange (FLEX) approximation. We find that the parquet approximation, where the fully irreducible vertex is approximated by the bare vertex, shows satisfactory agreement with DQMC and a significant improvement from FLEX or self-consistent second order approximation.