  Determining the ground state properties of the two-dimensional Hubbard model
has remained an outstanding problem. Applying recent advances in constrained
path auxiliary-field quantum Monte Carlo techniques and simulating large
rectangular periodic lattices, we calculate the long-range spin and charge
correlations in the ground state as a function of doping. At intermediate
interaction strengths, an incommensurate spin density wave (SDW) state is
found, with antiferromagnetic order and essentially homogeneous charge
correlation. The wavelength of the collective mode decreases with doping, as
does its magnitude. The SDW order vanishes beyond a critical doping. As the
interaction is increased, the holes go from a wave-like to a particle-like
state, and charge ordering develops which eventually evolves into stripe-like
states.
