  A zero-site density matrix renormalization algorithm (DMRG0) is proposed to
minimize the energy of matrix product states (MPS). Instead of the site tensors
themselves, we propose to optimize sequentially the "message" tensors between
neighbor sites, which contain the singular values of the bipartition. This
leads to a local minimization step that is independent of the physical
dimension of the site. Conceptually, it separates the optimization and
decimation steps in DMRG. Furthermore, we introduce two new global
perturbations based on the optimal low-rank correction to the current state,
which are used to avoid local minima. They are determined variationally as the
MPS closest to the one-step correction of the Lanczos or Jacobi-Davidson
eigensolver, respectively. These perturbations mainly decrease the energy and
are free of hand-tuned parameters. Compared to existing single-site enrichment
proposals, our approach gives similar convergence ratios per sweep while the
computations are cheaper by construction. Our methods may be useful in systems
with many physical degrees of freedom per lattice site. We test our approach on
the periodic Heisenberg spin chain for various spins, and on free electrons on
the lattice.
