  We present a comprehensive study on the extraction of CFT data using tensor
network methods, specially, from the fixed-point tensor of the linearized
tensor renormalization group (lTRG) for the 2D classical Ising model near the
critical temperature. Utilizing two different methods, we extract operator
scaling dimensions and operator-product-expansion (OPE) coefficients by
introducing defects on the lattice and by employing the fixed-point tensor. We
also explore the effects of point-like defects in the lattice on the
coarse-graining process. We find that there is a correspondence between
coarse-grained defect tensors and conformal states obtained from lTRG
fixed-point equation. We also analyze the capabilities and limitations of our
proposed coarse-graining scheme for tensor networks with point-like defects,
which includes graph independent local truncation (GILT) and higher-order
tensor renormalization group (HOTRG). Our results provide a better
understanding of the capacity and limitations of the tenor renormalization
group scheme in coarse-graining defect tensors, and we show that GILT+HOTRG can
be used to give accurate two- and four-point functions under specific
conditions. We also find that employing the minimal canonical form further
improves the stability of the RG flow.
