Inspired by the Chern-Simons effective theory description of symmetry protected topological (SPT) phases in two dimensions, we present a projective construction for many-body wave functions of SPT phases. Using this projective construction, we can systematically write down trial wave functions 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 ν_b₁=ν_b₂=1/2 per site on a 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 an emergent QED₃ with fermion number Nf=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.
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Lu et al. (2014) studied this question.