  We study the many-body phases of bosonic atoms with $N$ internal states
confined to a 1D optical lattice under the influence of a synthetic magnetic
field and strong repulsive interactions. The $N$ internal states of the atoms
are coupled via Raman transitions creating the synthetic magnetic field in the
space of internal spin states corresponding to recent experimental
realisations. We focus on the case of strong $\mbox{SU}(N)$ invariant local
density-density interactions in which each site of the 1D lattice is at most
singly occupied, and strong Raman coupling, in distinction to previous work
which has focused on the weak Raman coupling case. This allows us to keep only
a single state per site and derive a low energy effective spin $1/2$ model. The
effective model contains first-order nearest neighbour tunnelling terms, and
second-order nearest neighbour interactions and correlated next-nearest
neighbour tunnelling terms. By adjusting the flux $\phi$ one can tune the
relative importance of first-order and second-order terms in the effective
Hamiltonian. In particular, first-order terms can be set to zero, realising a
novel model with dominant second-order terms. We show that the resulting
competition between density-dependent tunnelling and repulsive density-density
interaction leads to an interesting phase diagram including a phase with
long-ranged pair-superfluid correlations. The method can be straightforwardly
extended to higher dimensions and lattices of arbitrary geometry including
geometrically frustrated lattices where the interplay of frustration,
interactions and kinetic terms is expected to lead to even richer physics.
