Exotic Superfluid with Emergent flux in a one-dimensional Bose-Fermi mixture

We find a novel chiral superfluid (CSF) phase in a one-dimensional Bose-Fermi Hubbard model with significant mass and density imbalance between the two species. In the CSF phase, bosons condensate at non-zero momentum $\pm 2\pi /L$ with chain length $L$. To capture the essential physics of this new phenomenon, we study an alternative simplified model that only features competition between single-fermion hopping and hopping of composite particles composed of a fermion and a boson. This model captures the low energy physics of the Hubbard model and hosts a robust CSF phase. Our unbiased numerical studies show that in the CSF phase, the local superfluid order parameter continuously rotates along the chain, indicating that time-reversal symmetry is spontaneously broken. This symmetry breaking generates an emergent flux in the background, effectively optimizing the system's ground-state energy. We provide a physical understanding at the mean-field level. Furthermore, we have explored the potential realization of this phase in cold-atom experiments.