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

TL;DR

This paper introduces a method combining tensor network renormalization and conformal field theory to analyze the RG flow of classical models, revealing finite bond dimension effects and emergent perturbations.

Contribution

It presents a novel approach to extract coupling constants and visualize RG flow in classical models, highlighting finite bond dimension limitations in TNR.

Findings

Finite correlation length due to finite bond dimension D in TNR.

Correlation length scales with D similarly to finite entanglement scaling.

Method successfully applied to Ising and 3-state Potts models.

Abstract

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 (RG) flow and demonstrate it with the classical Ising and 3-state Potts models. Furthermore, utilizing the new 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.