  The recently proposed Householder transformed density-matrix functional
embedding theory (Ht-DMFET) [Sekaran et al., Phys. Rev. B 104, 035121 (2021)],
which is equivalent to (but formally simpler than) density matrix embedding
theory (DMET) in the non-interacting case, is revisited from the perspective of
density-functional theory (DFT). An in-principle-exact density-functional
version of Ht-DMFET is derived for the one-dimensional Hubbard lattice with a
single embedded impurity. On the basis of well-identified density-functional
approximations, a local potential functional embedding theory (LPFET) is
formulated and implemented. Even though LPFET performs better than Ht-DMFET in
the low-density regime, in particular when electron correlation is strong, both
methods are unable to describe the density-driven Mott-Hubbard transition, as
expected. These results combined with our formally exact density-functional
embedding theory reveal that a single statically embedded impurity can in
principle describe the gap opening, provided that the complementary correlation
potential (that describes the interaction of the embedding cluster with its
environment, which is simply neglected in both Ht-DMFET and LPFET) exhibits a
derivative discontinuity (DD) at half filling. The extension of LPFET to
multiple impurities (which would enable to circumvent the modeling of DDs) and
its generalization to quantum chemical Hamiltonians are left for future work.
