  We show a way to perform the canonical renormalization group (RG)
prescription in tensor space: write down the tensor RG equation, linearize it
around a fixed-point tensor, and diagonalize the resulting linearized RG
equation to obtain scaling dimensions. The tensor RG methods have had a great
success in producing accurate free energy compared with the conventional
real-space RG schemes. However, the above-mentioned canonical procedure has not
been implemented for general tensor-network-based RG schemes. We extend the
success of the tensor methods further to extraction of scaling dimensions
through the canonical RG prescription, without explicitly using the conformal
field theory. This approach is benchmarked in the context of the Ising models
in 1D and 2D. Based on a pure RG argument, the proposed method has potential
applications to 3D systems, where the existing bread-and-butter method is
inapplicable.
