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We propose a general procedure for extracting the running coupling constantsof the underlying field theory of a given classical statistical model on atwo-dimensional lattice, combining tensor network renormalization (TNR) and thefinite-size scaling theory of conformal field theory. By tracking the couplingconstants at each scale, we are able to visualize the renormalization group(RG) flow and demonstrate it with the classical Ising and 3-state Potts models.Furthermore, utilizing the new methodology, we reveal the limitations due tofinite bond dimension D on TNR applied to critical systems. We find that afinite correlation length is imposed by the finite bond dimension in TNR, andit can be attributed to an emergent relevant perturbation that respects thesymmetries of the system. The correlation length shows the same power-lawdependence on D as the "finite entanglement scaling" of the Matrix ProductStates.

Finite-size and finite bond dimension effects of tensor network renormalization