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The internal degrees of freedom provided by ultracold atoms give a route forrealizing higher dimensional physics in systems with limited spatialdimensions. Non-spatial degrees of freedom in these systems are dubbed"synthetic dimensions". This connection is useful from an experimentalstandpoint but complicated by the fact that interactions alter the condensateground state. Here we use the Gross-Pitaevskii equation to study ground stateproperties of a spin-1 Bose gas under the combined influence of an opticallattice, spin-orbit coupling, and interactions at the mean field level. Theassociated phases depend on the sign of the spin-dependent interactionparameter and the strength of the optical lattice potential. We find "charge"and spin density wave phases which are directly related to helical spin orderin real space and affect the behavior of edge currents in the syntheticdimension. We determine the resulting phase diagram as a function of thespin-orbit coupling and spin-dependent interaction strength, considering bothattractive (ferromagnetic) and repulsive (polar) spin-dependent interactions.Our results are applicable to current and future experiments, specifically with$^{87}$Rb, $^{7}$Li, $^{41}$K, and $^{23}$Na.

Real space mean-field theory of a spin-1 Bose gas in synthetic dimensions