Matrix product approximations to multipoint functions in two-dimensional conformal field theory

TL;DR

This paper demonstrates that matrix product states can effectively approximate correlation functions in two-dimensional conformal field theories, providing explicit constructions and error estimates, thus bridging a gap in understanding their applicability to critical quantum systems.

Contribution

The authors provide the first rigorous quantitative estimates for MPS approximations of 2D conformal field theory correlations, including explicit constructions and error bounds.

Findings

MPS can approximate 2D CFT correlation functions with controlled error

Explicit MPS constructions are provided for these approximations

The work offers a rigorous foundation for variational methods in critical quantum systems

Abstract

Matrix product states (MPS) illustrate the suitability of tensor networks for the description of interacting many-body systems: ground states of gapped $1$-D systems are approximable by MPS as shown by Hastings [J. Stat. Mech. Theor. Exp., P08024 (2007)]. In contrast, whether MPS and more general tensor networks can accurately reproduce correlations in critical quantum systems, respectively quantum field theories, has not been established rigorously. Ample evidence exists: entropic considerations provide restrictions on the form of suitable Ansatz states, and numerical studies show that certain tensor networks can indeed approximate the associated correlation functions. Here we provide a complete positive answer to this question in the case of MPS and $2D$ conformal field theory: we give quantitative estimates for the approximation error when approximating correlation functions by MPS. Our work is constructive and yields an explicit MPS, thus providing both suitable initial values as well as a rigorous justification of variational methods.