Several variants of the recently proposed Density Matrix Embedding Theory
(DMET) [G. Knizia and G. K-L. Chan, Phys. Rev. Lett. 109, 186404 (2012)] are
formulated and tested. We show that spin symmetry breaking of the lattice
mean-field allows precise control of the lattice and fragment filling while
providing very good agreement between predicted properties and exact results.
We present a rigorous proof that at convergence this method is guaranteed to
preserve lattice and fragment filling. Differences arising from fitting the
fragment one-particle density matrix alone versus fitting fragment plus bath
are scrutinized. We argue that it is important to restrict the density matrix
fitting to solely the fragment. Furthermore, in the proposed broken symmetry
formalism, it is possible to substantially simplify the embedding procedure
without sacrificing its accuracy by resorting to density instead of density
matrix fitting. This simplified Density Embedding Theory (DET) greatly improves
the convergence properties of the algorithm
