Mott insulator-density ordered superfluid transition and‘shamrock transition’in a frustrated triangle lattice

Density order is usually a consequence of the competition between long-range and short-range interactions.Here we report a density ordered superfluid emergent from a homogeneous Mott insulator due to the competition between frustrations and local interactions.This transition is found in a Bose–Hubbard model on a frustrated triangle lattice with an extra pairing term.Furthermore,we find a quantum phase transition between two different density ordered superfluids,which is beyond the Landau–Ginzburg(LG)paradigm.A U(1)symmetry is emergent at the critical point,while the symmetry in each density ordered superfluid is Z_(2)×Z_(3).We call the transition a‘shamrock transition’,due to its degenerate ground state in the parameter space being a shamrock-like curve rather than a circle in an LG-type transition.Effective low energy theories are established for the two transitions mentioned above and we find their resemblance and differences with clock models.