  We study the renormalization group flow of the Lagrangian for statistical and
quantum systems by representing their path integral in terms of a tensor
network. Using a tensor-entanglement-filtering renormalization (TEFR) approach
that removes local entanglement and coarse grain the lattice, we show that the
resulting renormalization flow of the tensors in the tensor network has a nice
fixed-point structure. The isolated fixed-point tensors $T_{inv}$ plus the
symmetry group $G_{sym}$ of the tensors (i.e. the symmetry group of the
Lagrangian) characterize various phases of the system. Such a characterization
can describe both the symmetry breaking phases and topological phases, as
illustrated by 2D statistical Ising model, 2D statistical loop gas model, and
1+1D quantum spin-1/2 and spin-1 models. In particular, using such a $(G_{sym},
T_{inv}) $ characterization, we show that the Haldane phase for a spin-1 chain
is a phase protected by the time-reversal, parity, and translation symmetries.
Thus the Haldane phase is a symmetry protected topological phase. The
$(G_{sym}, T_{inv})$ characterization is more general than the
characterizations based on the boundary spins and string order parameters. The
tensor renormalization approach also allows us to study continuous phase
transitions between symmetry breaking phases and/or topological phases. The
scaling dimensions and the central charges for the critical points that
describe those continuous phase transitions can be calculated from the
fixed-point tensors at those critical points.
