Matrix Product Approximations to Multipoint Functions in Two-Dimensional Conformal Field Theory

Matrix product states (MPSs) illustrate the suitability of tensor networks for the description of interacting many-body systems: ground states of gapped 1D systems are approximable by MPSs, as shown by Hastings [M. B. Hastings, J. Stat. Mech. (2007) P08024]. By contrast, whether MPSs and more general tensor networks can accurately reproduce correlations in critical quantum systems or 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 MPSs and 2D conformal field theory: we give quantitative estimates for the approximation error when approximating correlation functions by MPSs. Our work is constructive and yields an explicit MPS, thus providing both suitable initial values and a rigorous justification of variational methods.

Figure 1

Graphical representation of a MPS. Maximally entangled pairs of dimension $D$ are placed on the bonds between physical $d$-dimensional quantum systems. The circles represent the $d\times D^2$ dimensional tensor given by the $D\times D$ matrices $A_1,\dots ,A_d$.

Figure 2

Illustration of the renormalization procedure. (Left panel) The regularized field operators ${\mathsf{W}}_q$ create arbitrary superpositions of energy eigenstates; hence, an infinite amount of entanglement is needed to implement their action. The renormalized field operators ${\mathsf{W}}_q^N$ (right panel: $N=3$) only create superpositions of, at most, $N$ different energy eigenstates if applied to one element of the basis. Yet their action can be shown to be approximately the same as ${\mathsf{W}}_q$.