  Using variational density matrix optimization with two- and three-index
conditions we study the one-dimensional Hubbard model with periodic boundary
conditions at various filling factors. Special attention is directed to the
full exploitation of the available symmetries, more specifically the
combination of translational invariance and space-inversion parity, which
allows for the study of large lattice sizes. We compare the computational
scaling of three different semidefinite programming algorithms with increasing
lattice size, and find the boundary point method to be the most suited for this
type of problem. Several physical properties, such as the two-particle
correlation functions, are extracted to check the physical content of the
variationally determined density matrix. It is found that the three-index
conditions are needed to correctly describe the full phase diagram of the
Hubbard model. We also show that even in the case of half filling, where the
ground-state energy is close to the exact value, other properties such as the
spin-correlation function can be flawed.
