We study two-dimensional bosonic symmetry-protected topological (SPT) phases that are protected by reflection symmetry and local symmetry [${Z}_{N}\ensuremath{\rtimes}R, {Z}_{N}\ifmmode\times\else\texttimes\fi{}R$, U(1)$\ensuremath{\rtimes}R$, or U(1)$\ifmmode\times\else\texttimes\fi{}R$], in the search for two-dimensional bosonic analogs of topological crystalline insulators in integer-
S spin systems with reflection and spin-rotation symmetries. To classify them, we employ a Chern-Simons approach and examine the stability of edge states against perturbations that preserve the assumed symmetries. We find that SPT phases protected by ${Z}_{N}\ensuremath{\rtimes}R$ symmetry are classified as ${\mathbb{Z}}_{2}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}$ for even
N and 0 (no SPT phase) for odd
N, while those protected by U(1)$\ensuremath{\rtimes}R$ symmetry are
Z2. We point out that the two-dimensional Affleck-Kennedy-Lieb-Tasaki state of
S=2 spins on the square lattice is a
Z2 SPT phase protected by reflection and $\ensuremath{\pi}$-rotation symmetries.
Symmetry-protected topological phases (SPT) are short-range entangled gapped states protected by global symmetry. Nontrivial SPT phases cannot be adiabatically connected to the trivial disordered state (or atomic insulator) as long as certain global symmetry
G is unbroken. At low energies, most of the two-dimensional SPTs with Abelian symmetry can be described by topological quantum field theory (TQFT) of the multicomponent Chern-Simons type. However, in contrast to the fractional quantum Hall effect where TQFT can give rise to interesting bulk anyons, TQFT for SPTs only supports trivial bulk excitations. The essential question in TQFT descriptions for SPTs is to understand how the global symmetry is implemented in the partition function. In this paper, we systematically study TQFT of three-dimensional SPTs with unitary Abelian symmetry (e.g., ${\mathbb{Z}}_{{N}_{1}}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{{N}_{2}}\ifmmode\times\else\texttimes\fi{}...$). In addition to the usual multicomponent
BF topological term at level-1, we find that there are new topological terms with quantized coefficients (e.g., ${a}^{1}\ensuremath{\wedge}{a}^{2}\ensuremath{\wedge}d{a}^{2}$ and ${a}^{1}\ensuremath{\wedge}{a}^{2}\ensuremath{\wedge}{a}^{3}\ensuremath{\wedge}{a}^{4}$) in TQFT actions, where
a1,a2,... are 1-form U(1) gauge fields. These additional topological terms cannot be adiabatically turned off as long as
G is unbroken. By investigating symmetry transformations for the TQFT partition function, we end up with the classification of SPTs that is consistent with the well-known group cohomology approach. We also discuss how to gauge the global symmetry and possible TQFT descriptions of Dijkgraaf-Witten gauge theory.
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 ${\ensuremath{\nu}}_{{b}_{1}}={\ensuremath{\nu}}_{{b}_{2}}=\frac{1}{2}$ 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.