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To explore superfluidity in flat-band systems, we consider a Bose-Hubbardmodel on a cross-linked ladder with $\pi$ flux, which has a flat band with agap between the other band for noninteracting particles, where we study theeffect of the on-site repulsion nonperturbatively. For low densities, we findexact degenerate ground states, each of which is a Wigner solid withnonoverlapping Wannier states on the flat band. At higher densities, themany-body system, when projected onto the lower flat band, can be mapped to aspin-chain model. This mapping enables us to reveal the existence of aTomonaga-Luttinger liquid comprising it pairs of bosons. Interestingly, thehigh- and low- density regions have an overlap, where the two phases coexist.

Phase diagram and pair Tomonaga-Luttinger liquid in a Bose-Hubbard model with flat bands