Formation of paired phases of bosons and their excitations in a square lattice

We investigate the formation of paired states of bosons in an optical lattice, namely, pair superfluid (PSF) and pair supersolid (PSS) in the presence of pair hopping as well as the next nearest neighbor (NNN) interaction mimicking long-range forces. Both the zero and finite temperature phase diagrams are obtained using the cluster mean field theory, which includes the effect of correlations systematically. We also compute the low-energy excitations which capture the characteristic features of such paired states and their transitions. Apart from the gapless sound mode due to the PSF order, a gapped mode also appears in the PSF phase, similar to the Higgs mode of the usual atomic superfluid (ASF). The PSF to ASF transition exhibits an intriguing behavior due to the existence of a `tri-critical' point, where the nature of transition changes. As a consequence of the continuous PSF-ASF transition, the gapped mode of both the phases becomes gapless at the critical point. For sufficiently strong NNN interaction strength, a PSS phase appears with coexisting pair superfluidity and stripe density order. The softening of the roton mode as a precursor of density ordering and the appearance of a low-energy gapped mode serve as robust features related to the formation of the PSS phase. We also investigate the melting of PSF and PSS phases to normal fluid at finite temperatures, particularly the melting pathway of PSS which occurs in atleast two steps due to the coexisting orders. Finally, we discuss the possibility of emulating such exotic phases in the ongoing cold atom experiments.