Coherent coupling generated by laser light between the hyperfine states of atoms, loaded in a 1D optical lattice, gives rise to the synthetic dimension system which is equivalent to a Hofstadter model in a finite strip of square lattice. An SU(M) symmetric attractive interaction in conjunction with the synthetic gauge field present in this system gives rise to unusual effects. We study the two- body problem of the system using the T-matrix formalism. We show that the two-body ground states pick up a finite momentum and can transform into two-body resonance like features in the scattering continuum with a large change in the phase shift. As a result, even for this 1D system, a critical amount of attraction is needed to form bound states. These phenomena have spectacular effects on the many body physics of the system analyzed using the numerical density matrix renormalization group technique. We show that the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) states form in the system even for a balanced gas and the FFLO momentum of the pairs scales linearly with flux. Considering suitable measures, we investigate interesting properties of these states. We also discuss a possibility of realization of a generalized interesting topological model, called the Creutz ladder.
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