For simplicity, we will use the one dimensional Hubbard model to illustrate the MSMB formalism. This low dimension is also the most difficult regime for quantum cluster approaches like the DCA. The Hubbard Hamiltonian is given by
with creating an electron of spin at site and local density . The first part, the kinetic term, allows hopping between adjacent lattice sites with transfer integral . The second term is the on-site Coulomb repulsion making a doubly occupied lattice site unfavorable. Throughout the remainder of this paper we choose the bare bandwidth as the unit of energy by setting and work at fixed filling .