Phases and dynamics of the one-dimensional interacting bosons with double-well dispersion role

Spin-orbit coupling as well as Floquet techniques can generate dispersion for one dimensional Bosons with a double well dispersion. This leads to a Z2 symmetry breaking Lifshitz quantum critical point and allows for a rich phase diagram for the interacting bosons, distinctive from the well-established Lieb-Liniger model. Using a combination of effective models as well as mapping to fermions, we construct a phase diagram and study the effect of quantum fluctuations generated by strong interactions on the phases as well as dynamical response in the vicinity of the quantum critical point. Specifically, we will discuss a phase with more than one gapless mode as well as an unusual domain wall dynamics in the Z2 symmetry broken phase.