Fixed-point tensor is a four-point function

TL;DR

This paper demonstrates how a fixed-point tensor derived from a 2D critical lattice model encodes the conformal field theory data, including four-point functions and operator product expansion coefficients, enabling complete CFT characterization.

Contribution

It introduces a method to extract full CFT data directly from the fixed-point tensor of a critical lattice model, linking tensor networks with conformal field theory.

Findings

Explicit fixed-point tensor elements match CFT four-point functions

Operator product expansion coefficients are obtained from tensor elements

Complete CFT data can be extracted from the tensor for any critical unitary lattice model

Abstract

Through coarse-graining, tensor network representations of a two-dimensional critical lattice model flow to a universal four-leg tensor, corresponding to a conformal field theory (CFT) fixed-point. We computed explicit elements of the critical fixed-point tensor, which we identify as the CFT four-point function. This allows us to directly extract the operator product expansion coefficients of the CFT from these tensor elements. Combined with the scaling dimensions obtained from the transfer matrix, we determine the complete set of the CFT data from the fixed-point tensor for any critical unitary lattice model.