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

We propose a general procedure for extracting the running coupling constants of the underlying field theory of a given classical statistical model on a two-dimensional lattice, combining tensor network renormalization (TNR) and the finite-size scaling theory of conformal field theory. By tracking the coupling constants at each scale, we are able to visualize the renormalization group flow and demonstrate it with the classical Ising and three-state Potts models. Furthermore, utilizing this methodology, we reveal the limitations due to finite bond dimension $D$ on TNR applied to critical systems. We find that a finite correlation length is imposed by the finite bond dimension in TNR, and it can be attributed to an emergent relevant perturbation that respects the symmetries of the system. The correlation length shows the same power-law dependence on $D$ as the ``finite entanglement scaling'' of the matrix product states.