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. Further, 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 $Z_2\times Z_3$. We call the transition ``shamrock transition" due to its degenerate ground state in parameter space being a shamrock-like curve rather than a circle in 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.