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Continuous Tensor Network States for Quantum Fields

Antoine Tilloy and J. Ignacio Cirac
Phys. Rev. X 9, 021040 – Published 28 May 2019
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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 to spatial dimensions d2. 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, we propose a generalization bearing similarities with the continuum multiscale entanglement renormalization ansatz.

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  • Received 3 August 2018
  • Revised 12 February 2019

DOI:https://doi.org/10.1103/PhysRevX.9.021040

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.

Published by the American Physical Society

Physics Subject Headings (PhySH)

Quantum Information, Science & TechnologyCondensed Matter, Materials & Applied PhysicsParticles & Fields

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Pushing Tensor Networks to the Limit

Published 28 May 2019

An extension of tensor networks—mathematical tools that simplify the study of complex quantum systems—could allow their application to a broad range of quantum field theory problems.

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Authors & Affiliations

Antoine Tilloy* and J. Ignacio Cirac

  • Max-Planck-Institut für Quantenoptik, Hans-Kopfermann-Straße 1, 85748 Garching, Germany

  • *antoine.tilloy@mpq.mpg.de
  • ignacio.cirac@mpq.mpg.de

Popular Summary

Matter composed of many quantum particles exhibits a rich diversity of behavior, but the numerous degrees of freedom make it challenging to capture all of this behavior mathematically. Tensor network states are mathematical tools that provide an “economical” description of such systems, allowing researchers to compute their properties and classify exotic phases of matter. However, the limitation of tensor networks is that the particles can appear only at discrete positions on a grid, and modifications that allow for freely positioned particles have been defined only for 1D systems. Here, we provide a way to utilize tensor networks in continuum systems with two or more dimensions.

Specifically, we put forward a class of states for quantum fields obtained as a continuum limit of tensor networks on a lattice with more than one dimension. We also show how the important properties of the tool—expressiveness, gauge invariance, scaling—translate to the continuum.

Our framework greatly extends the reach of tensor-network methods to study problems posed by continuous systems, allowing researchers to directly study various quantum field theories without the need for a prior coarse-graining of space. Reciprocally, it also allows researchers to import continuum techniques, hybridized with tensor networks, to approximately solve systems on a lattice.

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Issue

Vol. 9, Iss. 2 — April - June 2019

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