Quantum phase transitions between bosonic symmetry protected topological phases in two dimensions: emergent $QED_3$ and anyon superfluid

Inspired by Chern-Simons effective theory description of symmetry protected topological (SPT) phases in two dimensions, we present a projective construction for many-body wavefunctions of SPT phases. Using this projective construction we can systematically write down trial wavefunctions of SPT phases on a lattice. An explicit example of SPT phase with $U(1)$ symmetry is constructed for two types of bosons with filling $\nu_{b_1}=\nu_{b_2}=\frac12$ per site on square lattice. We study continuous phase transitions between different $U(1)$-SPT phases based on projective construction. The effective theory around the critical point is emergent $QED_3$ with fermion number $N_f=2$. Such a continuous phase transition however needs fine tuning, and in general there are intermediate phases between different $U(1)$-SPT phases. We show that such an intermediate phase has the same response as an anyon superconductor, and hence dub it "anyon superfluid". A schematic phase diagram of interacting bosons with $U(1)$ symmetry is depicted.