  We investigate the finite-density phase diagram of a non-abelian SU(2)
lattice gauge theory in (1+1)-dimensions using tensor network methods. We
numerically characterise the phase diagram as a function of the matter filling
and of the matter-field coupling, identifying different phases, some of them
appearing only at finite densities. For weak matter-field coupling we find a
meson BCS liquid phase, which is confirmed by second-order analytical
perturbation theory. At unit filling and for strong coupling, the system
undergoes a phase transition to a charge density wave of single-site (spin-0)
mesons via spontaneous chiral symmetry breaking. At finite densities, the
chiral symmetry is restored almost everywhere, and the meson BCS liquid becomes
a simple liquid at strong couplings, with the exception of filling two-thirds,
where a charge density wave of mesons spreading over neighbouring sites
appears. Finally, we identify two tri-critical points between the chiral and
the two liquid phases which are compatible with a $SU(2)_2$
Wess-Zumino-Novikov-Witten model. Here we do not perform the continuum limit
but we explicitly address the global $U(1)$ charge conservation symmetry.
