Matrix product approximations to conformal field theories

TL;DR

This paper provides rigorous bounds for approximating conformal field theory correlation functions using tensor networks, specifically matrix product states, with polynomial and sub-exponential scaling in key parameters.

Contribution

It introduces a novel method to approximate CFT correlation functions with finite-dimensional tensor networks and establishes error bounds with explicit scaling behaviors.

Findings

Bond dimension scales polynomially with inverse error

Approximation is sub-exponential in the ultraviolet cutoff

Group-covariant MPS correspond to the proposed approximation

Abstract

We establish rigorous error bounds for approximating correlation functions of conformal field theories (CFTs) by certain finite-dimensional tensor networks. For chiral CFTs, the approximation takes the form of a matrix product state. For full CFTs consisting of a chiral and an anti-chiral part, the approximation is given by a finitely correlated state. We show that the bond dimension scales polynomially in the inverse of the approximation error and sub-exponentially in the ultraviolett cutoff. We illustrate our findings using Wess-Zumino-Witten models, and show that there is a one-to-one correspondence between group-covariant MPS and our approximation.