Continuous Tensor Network States for Quantum Fields

TL;DR

This paper introduces a new class of continuum tensor network states for bosonic quantum fields, extending discrete tensor networks to higher dimensions with properties like Euclidean invariance and simple local observable expressions.

Contribution

It generalizes tensor network states to the continuum for quantum fields in higher dimensions, incorporating invariance and continuum limits, and proposes a new continuum multi-scale entanglement approach.

Findings

States are Euclidean invariant and continuum limits of discrete tensor networks.

They have simple rescaling flows and compact expressions for local observables.

The framework extends PEPS and suggests a continuum MERA generalization.

Abstract

We introduce a new class of states for bosonic quantum fields which extend tensor network states to the continuum and generalize continuous matrix product states (cMPS) to spatial dimensions $d\geq 2$. By construction, they are Euclidean invariant, and are genuine continuum limits of discrete tensor network states. Admitting both a functional integral and an operator representation, they share the important properties of their discrete counterparts: expressiveness, invariance under gauge transformations, simple rescaling flow, and compact expressions for the $N$-point functions of local observables. While we discuss mostly the continuous tensor network states extending Projected Entangled Pair States (PEPS), we propose a generalization bearing similarities with the continuum Multi-scale Entanglement Renormalization Ansatz (cMERA).