  We determine the spin and charge orders in the ground state of the doped
two-dimensional (2D) Hubbard model in its simplest form, namely with only
nearest-neighbor hopping and on-site repulsion. At half-filling, the ground
state is known to be an anti-ferromagnetic Mott insulator. Doping Mott
insulators is believed to be relevant to the superconductivity observed in
cuprates. A variety of candidates have been proposed for the ground state of
the doped 2D Hubbard model. A recent work employing a combination of several
state-of-the-art numerical many-body methods, established the stripe order as
the ground state near $1/8$ doping at strong interactions. In this work, we
apply one of these methods, the cutting-edge constrained-path auxiliary field
quantum Monte Carlo method with self-consistently optimized gauge constraints,
to systematically study the model as a function of doping and interaction
strength. With careful finite size scaling based on large-scale computations,
we map out the ground state phase diagram in terms of its spin and charge
order. We find that modulated antiferromagnetic order persists from near
half-filling to about $1/5$ doping. At lower interaction strengths or larger
doping, these ordered states are best described as spin-density waves, with
essentially delocalized holes and modest oscillations in charge correlations.
When the charge correlations are stronger (large interaction or small doping),
they are best described as stripe states, with the holes more localized near
the node in the antiferromagnetic spin order. In both cases, we find that the
wavelength in the charge correlations is consistent with so-called filled
stripes in the pure Hubbard model.
