Matrix product approximations to conformal field theories

Robert König,Volkher B Scholz

doi:10.1016/j.nuclphysb.2017.04.006

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 inverse of the minimal distance between insertion points. 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.

Highlights

Quantum field theory is arguably one of the most versatile physical theories developed to date
The chiral conformal field theories (CFTs) is defined in terms of a vertex operator algebras (VOAs) V and its modules, whereas the full CFT is defined in terms of a pair of VOAs
We further restrict ourselves to the case of unitary VOAs and modules, those possessing a scalar product turning them into a Hilbert space

Summary

Introduction

Quantum field theory is arguably one of the most versatile physical theories developed to date. The language of quantum field theory provides a sophisticated, unifying conceptual framework for addressing a variety of questions of physical interest. It constitutes one of the main pillars of modern physics. We again restrict our attention to the complex plane as well as the torus, where a rigorous construction of CFTs has been given by Huang and Kong [35,67] in terms an algebraic object called a conformal full field algebra. The algebraic construction of CFTs on highergenus surfaces remains a topic of ongoing research, but see the work of Fuchs, Runkel and Schweigert [77]. In the statistical mechanics (Euclidean) case we will choose zto be the complex conjugate of z, whereas for relativistic theories, both parameters stay independent, but real and imaginary part are equal to the lightcone variables before compactifying the space–time

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Conclusion