Continuous Transition between Bosonic Fractional Chern Insulator and Superfluid

The properties of fractional Chern insulator (FCI) phase and the phase transitions between FCI and Mott insulators (MI) in bosonic systems are well studied. The continuous transitions between FCI and superfluid (SF), however, despite the inspiring field theoretical predictions, have not been directly verified. The existing numerical results of FCI-SF transition are either indirect or clearly first-order. Here, by simply tuning the bandwidth of Haldane honeycomb lattice model, we find direct transitions from a bosonic FCI at $\nu$ = 1/2 filling of a flat Chern band to two SF states with bosons condensed at momenta $M$ and $\Gamma$, respectively. While the FCI-SF($M$) transition is first-order, the FCI-SF($\Gamma$) transition is found continuous. Through finite size criticality analysis, the obtained critical exponents $\beta\approx$0.35(5) and $\nu\approx$0.62(12) are both compatible with those of the 3D XY universality class and more exotic beyond-Landau ones. Our work thence presents a direct numerical evidence of a continuous FCI-SF transition between topological ordered phase and spontaneous continuous symmetry-breaking phase, and further indicates the zero-field bosonic FCI might be realized from a SF state by gradually flattening the dispersion of Chern band, through the (quasi)adiabatic preparation in ultracold atom systems.