Local scale transformations on the lattice with tensor network renormalization

TL;DR

This paper demonstrates how tensor network renormalization can implement local scale transformations on lattice models, enabling the extraction of conformal data such as scaling dimensions and OPE coefficients from critical systems.

Contribution

It introduces a method to perform lattice conformal maps using TNR, allowing direct access to conformal field theory data from lattice models.

Findings

Successfully applied to the 2D critical Ising model

Built a lattice version of the conformal map transforming plane to cylinder

Extracted scaling dimensions and OPE coefficients from the lattice

Abstract

Consider the partition function of a classical system in two spatial dimensions, or the Euclidean path integral of a quantum system in two space-time dimensions, both on a lattice. We show that the tensor network renormalization (TNR) algorithm [\emph{G. Evenbly and G. Vidal, Phys. Rev. Lett. 115, 180405}] can be used to implement local scale transformations on these objects, namely a lattice version of conformal maps. Specifically, we explain how to implement the lattice equivalent of the logarithmic conformal map that transforms the Euclidean plane into a cylinder. As an application, and with the 2D critical Ising model as a concrete example, we use this map to build a lattice version of the scaling operators of the underlying conformal field theory, from which one can extract their scaling dimensions and operator product expansion coefficients.