Volume 964

Lecture Notes in Physics

Series Editors

Matthias Bartelmann

Heidelberg, Germany

Roberta Citro

Salerno, Italy

Peter Hänggi

Augsburg, Germany

Morten Hjorth-Jensen

Oslo, Norway

Maciej Lewenstein

Barcelona, Spain

Angel Rubio

Hamburg, Germany

Manfred Salmhofer

Heidelberg, Germany

Wolfgang Schleich

Ulm, Germany

Stefan Theisen

Potsdam, Germany

James D. Wells

Ann Arbor, MI, USA

Gary P. Zank

Huntsville, AL, USA

Founding Editors

Wolf Beiglböck

Heidelberg, Germany

Jürgen Ehlers

Potsdam, Germany

Klaus Hepp

Zürich, Switzerland

Hans-Arwed Weidenmüller

Heidelberg, Germany

The Lecture Notes in Physics

The series Lecture Notes in Physics (LNP), founded in 1969, reports new developments in physics research and teaching-quickly and informally, but with a high quality and the explicit aim to summarize and communicate current knowledge in an accessible way. Books published in this series are conceived as bridging material between advanced graduate textbooks and the forefront of research and to serve three purposes:

• to be a compact and modern up-to-date source of reference on a well-defined topic

• to serve as an accessible introduction to the field to postgraduate students and nonspecialist researchers from related areas

• to be a source of advanced teaching material for specialized seminars, courses and schools

Both monographs and multi-author volumes will be considered for publication. Edited volumes should however consist of a very limited number of contributions only. Proceedings will not be considered for LNP.

Volumes published in LNP are disseminated both in print and in electronic formats, the electronic archive being available at springerlink.com. The series content is indexed, abstracted and referenced by many abstracting and information services, bibliographic networks, subscription agencies, library networks, and consortia.

Proposals should be sent to a member of the Editorial Board, or directly to the managing editor at Springer:

Dr Lisa Scalone

Springer Nature

Physics Editorial Department I

Tiergartenstrasse 17

69121 Heidelberg, Germany

lisa.scalone@springernature.com

More information about this series at http://www.springer.com/series/5304

Shi-Ju Ran, Emanuele Tirrito, Cheng Peng, Xi Chen, Luca Tagliacozzo, Gang Su and Maciej Lewenstein

Tensor Network Contractions

Methods and Applications to Quantum Many-Body Systems

../images/489509_1_En_BookFrontmatter_Figa_HTML.png

Shi-Ju Ran

Department of Physics, Capital Normal University, Beijing, China

Emanuele Tirrito

Quantum Optics Theory, Institute of Photonic Sciences, Castelldefels, Spain

Cheng Peng

Stanford Institute for Materials and Energy Sciences, SLAC and Stanford University, Menlo Park, CA, USA

Xi Chen

School of Physical Sciences, University of Chinese Academy of Science, Beijing, China

Luca Tagliacozzo

Department of Quantum Physics and Astrophysics, University of Barcelona, Barcelona, Spain

Gang Su

Kavli Institute for Theoretical Sciences, University of Chinese Academy of Science, Beijing, China

Maciej Lewenstein

Quantum Optics Theory, Institute of Photonic Sciences, Castelldefels, Spain

ISSN 0075-8450e-ISSN 1616-6361

Lecture Notes in Physics

ISBN 978-3-030-34488-7e-ISBN 978-3-030-34489-4

https://doi.org/10.1007/978-3-030-34489-4

This book is an open access publication.

Open Access This book is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.

The images or other third party material in this book are included in the book's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the book's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.

The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use.

The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This Springer imprint is published by the registered company Springer Nature Switzerland AG

The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Preface

Tensor network (TN), a young mathematical tool of high vitality and great potential, has been undergoing extremely rapid developments in the last two decades, gaining tremendous success in condensed matter physics, atomic physics, quantum information science, statistical physics, and so on. In this lecture notes, we focus on the contraction algorithms of TN as well as some of the applications to the simulations of quantum many-body systems. Starting from basic concepts and definitions, we first explain the relations between TN and physical problems, including the TN representations of classical partition functions, quantum many-body states (by matrix product state, tree TN, and projected entangled pair state), time evolution simulations, etc. These problems, which are challenging to solve, can be transformed to TN contraction problems. We present then several paradigm algorithms based on the ideas of the numerical renormalization group and/or boundary states, including density matrix renormalization group, time-evolving block decimation, coarse-graining/corner tensor renormalization group, and several distinguished variational algorithms. Finally, we revisit the TN approaches from the perspective of multi-linear algebra (also known as tensor algebra or tensor decompositions) and quantum simulation. Despite the apparent differences in the ideas and strategies of different TN algorithms, we aim at revealing the underlying relations and resemblances in order to present a systematic picture to understand the TN contraction approaches.

Shi-Ju Ran

Emanuele Tirrito

Cheng Peng

Xi Chen

Luca Tagliacozzo

Gang Su

Maciej Lewenstein

Beijing, ChinaCastelldefels, SpainMenlo Park, CA, USABeijing, ChinaBarcelona, SpainBeijing, ChinaCastelldefels, Spain

Acknowledgements

We are indebted to Mari-Carmen Bañuls, Ignacio Cirac, Jan von Delft, Yichen Huang, Karl Jansen, José Ignacio Latorre, Michael Lubasch, Wei Li, Simone Montagero, Tomotoshi Nishino, Roman Orús, Didier Poilblanc, Guifre Vidal, Andreas Weichselbaum, Tao Xiang, and Xin Yan for helpful discussions and suggestions. SJR acknowledges Fundació Catalunya-La Pedrera, Ignacio Cirac Program Chair and Beijing Natural Science Foundation (Grants No. 1192005). ET and ML acknowledge the Spanish Ministry MINECO (National Plan 15 Grant: FISICATEAMO No. FIS2016-79508-P, SEVERO OCHOA No. SEV-2015-0522, FPI), European Social Fund, Fundació Cellex, Generalitat de Catalunya (AGAUR Grant No. 2017 SGR 1341 and CERCA/Program), ERC AdG OSYRIS and NOQIA, EU FETPRO QUIC, and the National Science Centre, Poland-Symfonia Grant No. 2016/20/W/ST4/00314. LT was supported by the Spanish RYC-2016-20594 program from MINECO. SJR, CP, XC, and GS were supported by the NSFC (Grant No. 11834014). CP, XC, and GS were supported in part by the National Key R&D Program of China (Grant No. 2018FYA0305800), the Strategic Priority Research Program of CAS (Grant No. XDB28000000), and Beijing Municipal Science and Technology Commission (Grant No. Z118100004218001).

Acronyms