Fate of chiral order and impurity self-pinning in flat bands with local symmetry

Interacting bosons on a single plaquette threaded by a π flux can spontaneously break time-reversal symmetry, resulting in a chiral loop current. Connecting such bosonic π-flux plaquettes in a dispersive configuration was recently shown to lead to long-range chiral order. Here, instead, we design a chain of π-flux plaquettes that exhibits an all-flat-bands single-particle energy spectrum and an extensive set of local symmetries. Using Elitzur's theorem, we show that these local symmetries prevent the emergence of long-range chiral order. Moreover, projecting the dynamics to a Creutz ladder model with an effective intrarung interaction allows one to derive simple spin Hamiltonians capturing the ground-state degeneracy and the low-energy excitations, and to confirm the absence of chiral order. Nevertheless, we show how to obtain gauge-invariant information from a mean-field approach, which explicitly breaks gauge-invariance. Finally, we observe an impurity self-pinning phenomenon, when an extra boson is added on top of a ground state at integer filling, resulting in a nondispersive density peak. Exact diagonalization benchmarks are also provided, and experimental perspectives are discussed.