Field tensor network states

Anne E B Nielsen,Benedikt Herwerth,J Ignacio Cirac,Germán Sierra

doi:10.1103/physrevb.103.155130

Abstract

We define a class of tensor network states for spin systems where the individual tensors are functionals of fields. The construction is based on the path integral representation of correlators of operators in quantum field theory. These tensor network states are infinite dimensional versions of matrix product states and projected entangled pair states. We find the field-tensor that generates the Haldane-Shastry wave function and extend it to two dimensions. We give evidence that the latter underlies the topological chiral state described by the Kalmeyer-Laughlin wave function.

Highlights

Tensor networks (TNs) are becoming a key tool to describe many-body quantum systems [1]
We introduced a class of TN constructed using functionals of fields that are contracted by means of the path integral of the functions defined on the links of the network
We illustrate our approach using a massless boson in two dimensions that allows us to derive the Haldane-Shastry wave function that describes a critical state in the universality class given by the WZW model SU (2)1

Summary

INTRODUCTION

Tensor networks (TNs) are becoming a key tool to describe many-body quantum systems [1]. By simple we mean they have a small bond dimension, D, which limits the number of coefficients describing the tensors generating the many-body states The description of such states in terms of TNs automatically opens up the possibility of using powerful tools in order to describe their physical properties by just inspecting a simple tensor. The arguments above do not prevent the existence of exact descriptions of critical or chiral topological states with TNs of infinite bond dimensions. We take advantage of the fact that the Haldane-Shastry state [28,29], a prominent critical state, can be expressed in that form [24,30] to compute a FTN generating that state The description allows both periodic and open boundary conditions. We propose a FTN in two dimensions and give strong evidence that it represents a Kalmeyer-Laughlin state [12], a prototypical representative of chiral topological order

FTN IN ONE DIMENSION

L2 cosh2 t 2L

FTN IN TWO DIMENSIONS

CONCLUSIONS