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

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.

5 pages, 1 figure. See long companion paper arXiv:1509.07414 for full technical details

References in corpus (8)

• Matrix product states represent ground states faithfully
• Entropy scaling and simulability by Matrix Product States
• Scaling of entanglement support for Matrix Product States
• Continuous Matrix Product States for Quantum Fields
• Matrix product states for critical spin chains: finite size scaling versus finite entanglement scaling
• Holographic quantum states
• The renormalisation group via statistical inference
• Bosonic fractional quantum Hall states on the torus from conformal field theory

Cited by in corpus (5)

• Rigorous free fermion entanglement renormalization from wavelet theory
• Quantum circuit approximations and entanglement renormalization for the Dirac field in 1+1 dimensions
• Conformal field theory from lattice fermions
• Continuum limits of homogeneous binary trees and the Thompson group
• Exact bosonic Matrix Product States (and holography)