741 Analytic Trends in Mathematical Physics Arizona School of Analysis and Mathematical Physics March 5–9, 2018 University of Arizona, Tucson, Arizona Houssam Abdul-Rahman Robert Sims Amanda Young Editors Analytic Trends in Mathematical Physics Arizona School of Analysis and Mathematical Physics March 5–9, 2018 University of Arizona, Tucson, Arizona Houssam Abdul-Rahman Robert Sims Amanda Young Editors 741 Analytic Trends in Mathematical Physics Arizona School of Analysis and Mathematical Physics March 5–9, 2018 University of Arizona, Tucson, Arizona Houssam Abdul-Rahman Robert Sims Amanda Young Editors EDITORIAL COMMITTEE Dennis DeTurck, Managing Editor Michael Loss Kailash Misra Catherine Yan 2010 Mathematics Subject Classification. Primary 05C80, 60H15, 60J10, 81Q10, 81Q15, 82B20, 82B44, 82C10, 82C20, 82C22, 90B15. Library of Congress Cataloging-in-Publication Data Names: Abdul-Rahman, Houssam M.A., editor. | Sims, Robert, 1975- editor. | Young, Amanda, 1986- editor. | Arizona School of Analysis and Mathematical Physics (2018 : Tucson, Ariz.) Title: Analytic trends in mathematical physics / Houssam Abdul-Rahman, Robert Sims, Amanda Young, editors. Description: Providence, Rhode Island : American Mathematical Society, [2020] | Series: Con- temporary mathematics ; volume 741 | “Arizona School of Analysis and Mathematical Physics, March 5-9, 2018, University of Arizona, Tucson, Arizona.” | Includes bibliographical references. Identifiers: LCCN 2019028744 | ISBN 9781470448417 (paperback) — ISBN 9781470453886 (ebook) Subjects: LCSH: Mathematical physics. | Mathematical analysis. Classification: LCC QC20.7.A5 A54 2020 | DDC 530.155–dc23 LC record available at https://lccn.loc.gov/2019028744 DOI: https://doi.org/10.1090/conm/741 Color graphic policy. 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Visit the AMS home page at https://www.ams.org/ 10987654321 252423222120 Contents Preface vii List of participants ix A class of two-dimensional AKLT models with a gap Houssam Abdul-Rahman, Marius Lemm, Angelo Lucia, Bruno Nachtergaele, and Amanda Young 1 Note on linear response for interacting Hall insulators Sven Bachmann, Alex Bols, Wojciech De Roeck, and Martin Fraas 23 The adiabatic theorem in a quantum many-body setting Sven Bachmann, Wojciech De Roeck, and Martin Fraas 43 Perspectives on exponential random graphs Ryan DeMuse and Mei Yin 59 ASchr¨odinger operator approach to higher spin XXZ systems on general graphs Christoph Fischbacher 83 An index theorem for Schr¨odinger operators on metric graphs Yuri Latushkin and Selim Sukhtaiev 105 Finite-size criteria for spectral gaps in D-dimensional quantum spin systems Marius Lemm 121 The KPZ universality class and related topics Axel Saenz 133 Aspects of the mathematical theory of disordered quantum spin chains Gunter¨ Stolz 163 v Preface Successful researchers in mathematical physics often possess an informed ex- posure to the tools of mathematics and intuition derived from the methods of physics. A main goal of the Arizona School of Analysis and Mathematical Physics was to provide an environment where junior researchers could further develop in these directions. We organized speakers who would introduce current topics of ex- citing research that are both influenced by physical intuition and require the use of cutting-edge analytic techniques. Four mini-courses were designed to rigorously develop these important topics and illustrate how theory is used in problem solving. Given the success of the previous Arizona schools in 2009, 2010, and 2012, we were encouraged to continue this tradition by organizing another Arizona School of Analysis and Mathematical Physics, which took place at the University of Ari- zona from March 5–9, 2018. Mini-courses comprised of four one-hour lectures were given by Sven Bachmann (Adiabatic Dynamics and Linear Response Theory for Many-Body Quantum Systems), Fernando Brand˜ao (Entanglement Theory), Jan-Philip Solovej (Quasi-Free Variational Models for Bosons and Fermions), and G¨unter Stolz (Disordered Quantum Spin Chains). A number of senior participants (Bruno Nachtergaele, Jeff Schenker, Anna Vershynina, and Mei Yin) gave lectures on topics related to the main focus of the mini-courses, and a large fraction of the junior participants also gave short talks on their current research endeavors. In all, the diversity of participation was a testament to the enthusiasm this rich field generates. The works in this volume reflect recent progress and innovative techniques developed within mathematical physics. New results investigating spectral gaps are contained in the article by Abdul-Rahman, Lemm, Lucia, Nachtergaele, and Young as well as the one by Lemm. Recent applications exploring quantum dynamics can be found in the articles by Bachmann, Bols, De Roeck, and Fraas and also by Bachmann, De Roeck, and Fraas. Perspectives from the theory of Schr¨odinger operators motivate the articles by Fischbacher and by Latushkin and Sukhtaiev. Probabilistic applications to random models inspired the articles by DeMuse and Yin, by Saenz, and by Stolz. The works by Bachmann, De Roeck, and Fraas and by Stolz form detailed lecture notes for two of the school’s mini-courses. Moreover, the work by Abdul-Rahman, Lemm, Lucia, Nachtergaele, and Young is the product of discussions that emerged during a follow-up meeting to this Arizona School. We would like to give our thanks to the people and organizations that helped us with this school. Our local organizers Alejandra Gaona, David Gonzalez, and vii viii PREFACE Aubrey Mouradian were instrumental in all aspects concerning coordination. With- out them, none of this would have been possible. The bulk of the financial sup- port for this school was provided by the National Science Foundation (grant DMS- 1800724) to which we are extremely grateful. We also happily acknowledge ad- ditional support from the American Institute of Physics as well as crucial match- ing funds from the Department of Mathematics and the College of Science at the University of Arizona. These contributions were not only appreciated but also immensely important. Finally, our heart-felt thanks goes to all the lecturers, par- ticipants, and authors of the articles in this proceedings. Your contributions have made this school what it is: a great success! June 2019 Houssam Abdul-Rahman, Robert Sims, and Amanda Young List of participants H. Abdul-Rahman C. Fischbacher University of Arizona University of Alabama at Birmingham S. Bachmann S. Gamble University of British Columbia Virginia Tech S. Becker R. Ghanta University of Cambridge Georgia Institute of Technology A. Bols B. Hinrichs KU Leuven Friedrich-Schiller University of Jena F. Brand˜ao T. Hudgins California Institute of Technology Baylor University B. Brodie M. Jex University of Kentucky Karlsruhe Institute of Technology M. Cha F. Jiang Michigan State University University of California, Davis Y. Dai T. Kieffer University of Arizona Georgia Institute of Technology M. DarAssi J. Koberstein Princess Sumaya University for Friedrich-Schiller University of Jena Technology M. Lange M. Darras Karlsruhe Institute of Technology University of Alabama at Birmingham M. Lemm Harvard/IAS R. DeMuse University of Denver B. Li Case Western Reserve University A. Dietlein Ludwig-Maximilian University of M. Lukic Munich Rice University T. Easlick R. Matos University of Denver Michigan State University L. Emmert L. Meng Ludwig-Maximilian University of CEREMADE, Paris Dauphine Munich University ix x LIST OF PARTICIPANTS A. Moon A. Triay University of California, Davis CEREMADE, Paris Dauphine University B. Nachtergaele University of California, Davis A. Vershynina University of Houston S. Park University of Illinois M. Yin Urbana-Champaign University of Denver M. Phizicky A. Young University of Arizona University of Arizona R. Raquepas J. Zeng University Grenoble Alpes/McGill University of Oklahoma University S. Zhang J. Reschke Michigan State University University of California, Davis E. Sabo Georgia Institute of Technology A. Saenz University of Virginia K. Scarbrough University of Oklahoma J. Schenker