  Using the Density Matrix Renormalization Group technique we study the effect
of spin-orbit coupling on a three-orbital Hubbard model in the $(t_{2g})^{4}$
sector and in one dimension. Fixing the Hund coupling to a robust value
compatible with some multiorbital materials, we present the phase diagram
varying the Hubbard $U$ and spin-orbit coupling $\lambda$, at zero temperature.
Our results are shown to be qualitatively similar to those recently reported
using the Dynamical Mean Field Theory in higher dimensions, providing a robust
basis to approximate many-body techniques. Among many results, we observe an
interesting transition from an orbital-selective Mott phase to an excitonic
insulator with increasing $\lambda$ at intermediate $U$. In the strong $U$
coupling limit, we find a non-magnetic insulator with an effective angular
momentum $\langle(\textbf{J}^{eff})^{2}\rangle \ne 0$ near the excitonic phase,
smoothly connected to the $\langle(\textbf{J}^{eff})^{2}\rangle = 0$ regime. We
also provide a list of quasi-one dimensional materials where the physics
discussed in this publication could be realized.
