  We study the phase diagram and the total polarization distribution of the
Su-Schrieffer-Heeger model with nearest neighbor interaction in one dimension
at half-filling. To obtain the ground state wave-function, we extend the
Baeriswyl variational wave function to account for alternating hopping
parameters. The ground state energies of the variational wave functions compare
well to exact diagonalization results. For the case of uniform hopping for all
bonds, where it is known that an ideal conductor to insulator transition takes
place at finite interaction, we also find a transition at an interaction
strength somewhat lower than the known value. The ideal conductor phase is a
Fermi sea. The phase diagram in the whole parameter range shows a resemblance
to the phase diagram of the Kane-Mele-Hubbard model. We also calculate the
gauge invariant cumulants corresponding to the polarization (Zak phase) and use
these to reconstruct the distribution of the polarization. We calculate the
reconstructed polarization distribution along a path in parameter space which
connects two points with opposite polarization in two ways. In one case we
cross the metallic phase line, in the other, we go through only insulating
states. In the former case, the average polarization changes discontinuously
after passing through the metallic phase line, while in the latter the
distribution "walks across" smoothly from one polarization to its opposite.
This state of affairs suggests that the correlation acts to break the chiral
symmetry of the Su-Schrieffer-Heeger model, in the same way as it happens when
a Rice-Mele onsite potential is turned on.
