  In this paper, we provide a comprehensive study of the quantum magnetism in
the Mott insulating phases of the 1D Bose-Hubbard model with abelian or
non-abelian synthetic gauge fields, using the Density Matrix Renormalization
Group (DMRG) method. We focus on the interplay between the synthetic gauge
field and the asymmetry of the interactions, which give rise to a very general
effective magnetic model: a XYZ model with various Dzyaloshinskii-Moriya (DM)
interactions. The properties of the different quantum magnetic phases and
phases transitions of this model are investigated.
