Synthetic dimensions in the strong-coupling limit: Supersolids and pair superfluids

We study the many-body phases of bosonic atoms with $\small \textit{N}$ internal states confined to a one-dimensional (1D) optical lattice under the influence of a synthetic magnetic field and strong repulsive interactions. The $\small \textit{N}$ 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 realizations. We focus on the case of strong SU($\small \textit{N}$) 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-neighbor tunneling terms, second-order nearest-neighbor interactions, and correlated next-nearest-neighbor tunneling 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, realizing a model with dominant second-order terms. We show that the resulting competition between density-dependent tunneling and repulsive density-density interaction leads to an interesting phase diagram including a phase with long-range 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.Engineering and Physical Sciences Research Council (Grant ID: EP/K030094/1)This is the author accepted manuscript. The final version is available from the American Physical Society via http://dx.doi.org/10.1103/PhysRevA.94.02363