  We consider a modification of the one-dimensional Hubbard model by including
an external pairing potential. Guided by analytic bosonization results, we
quantitatively determine the grand-canonical zero-temperature phase diagram
using both finite and infinite density matrix renormalization group algorithm
based on the formalism of matrix product states and matrix product operator,
respectively. By computing various local quantities as well as the half-system
entanglement, we are able to distinguish between Mott, metallic and
superconducting phases. We point out the compressible nature of the Mott phase
and the fully gapped nature of the many-body spectrum of the superconducting
phase, in the presence of explicit U(1)-charge symmetry breaking.
