  Motivated by the nickel valence controversy in the perovskite nickelate
RNiO_3, we consider a one-dimensional underscreened Kondo chain consisting of
alternating spin-1 ("nickel") and electron ("oxygen") sites, which in addition
to the usual electron hopping and spin-spin interaction between the S=1 spin
and the electron also contains a spin mediated electron hopping term. Using the
density-matrix renormalization group (DMRG), we obtained the zero temperature
phase diagram of the model, as well as various correlation functions in each
phase. Importantly, for a certain range of parameters the model exhibits a
quasi-long-range spiral (QS) order. To understand the DMRG results, we
construct a mean-field theory based on a Schwinger fermion decomposition of the
S=1 spins, from which we argue that the QS phase corresponds to a phase in
proximity to the spin Bose metal state proposed by D. N. Sheng, O. I.
Motrunich, and M. P. A. Fisher [Phys. Rev. B 79, 205112 (2009)]. Notably, we
find no evidence for a phase with the symmetry of "nickel"-centered charge
order, which has been argued to arise due to site-selective Kondo screening of
half the S=1 spins, and suggest that order of this type occurs only due to an
additional energy gain from spontaneous lattice distortions, not present in
this model.
