Quantum phase transitions in condensed matter

Transcription

1 Quantum phase transitions in condensed matter Session L04: Lars Onsager Prize APS March Meeting, Los Angeles Subir Sachdev March 7, 2018 HARVARD Talk online: sachdev.physics.harvard.edu

2 Thanks to students and postdocs, and many other collatorators Jinwu Ye, Associate Professor, Department of Physics and Astronomy, Mississippi State University Thesis: Some Examples of Quantum Phase Transitions T. Senthil, Professor, Department of Physics, Massachusetts Institute of Technology. Thesis: Quantum Phase Transitions in Random Spin Systems Kedar Damle, Department of Theoretical Physics, Tata Institute of Fundamental Research, Mumbai, India. Thesis: Turning on the Heat:Non-zero Temperature Dynamical Properties of Quantum Many-body Systems Chiranjeeb Buragohain, Microsoft Research. Thesis: Dynamical Properties of Quantum Antiferromagnets in One and Two Dimensions Ying Zhang, Finisterre Capital, London. Thesis: Competing Orders in the Cuprate Superconductors Anatoli Polkovnikov, Associate Professor, Boston University. Thesis: Manifestation of Quantum Fluctuations in Strongly Correlated Systems Stephen Powell, Assistant Professor, University of Nottingham Thesis: Quantum phases and transitions of many-body systems realized using cold atomic gases Adrian Del Maestro, Associate Professor, University of Vermont Thesis: The superconductor-metal quantum phase transition in ultra-narrow wires Emily Dunkel (with David Coker, Boston University), NASA Jet Propulsion Laboratory Thesis: Quantum Phenomena in Condensed Phase Systems Yang Qi, Institute for Advanced Studies, Tsinghua University Thesis: Spin and Charge Fluctuations in Strongly Correlated Systems. Rudro Rana Biswas, Assistant Professor, Purdue University Thesis: Explorations in Dirac Fermions and Spin Liquids. Eun Gook Moon, Assistant Professor, Korea Advanced Institute of Science and Technology Thesis: Superfluidity in Strongly Correlated Systems Max Metlitski, Assistant Professor, Department of Physics, Massachusetts Institute of Technology Thesis: Aspects of Critical Behavior of Two Dimensional Electron Systems Yejin Huh, Applied Scientist at Apple Thesis: Quantum Phase Transitions in d-wave Superconductors and Antiferromagnetic Kagome Lattices Susanne Pielawa, Lyft, Munich Thesis: Metastable Phases and Dynamics of Low-Dimensional Strongly-Correlated Atomic Quantum Gases Debanjan Chowdhury, Moore Foundation Postdoctoral Fellow, MIT Thesis: Interplay of Broken Symmetries and Quantum Criticality in Correlated Electronic Systems Junhyun Lee, Postdoctoral fellow, University of Maryland Thesis: Novel quantum phase transitions in low-dimensional systems Andrew Lucas, Postdoctoral fellow, Stanford University Thesis: Transport and hydrodynamics in holography, strange metals and graphene Shubhayu Chatterjee, Harvard University Aavishkar Patel, Harvard University Wenbo Fu, Harvard University Seth Whitsitt, Harvard University Alex Thomson, Harvard University Julia Steinberg, Harvard University Students

3 Thanks to students and postdocs, and many other collatorators Pierre Le Doussal, Directeur de Recherche de Classe Exceptionnelle, Laboratoire de Physique Théorique de l' Ecole Normale Supérieure, Paris, France. Rodolfo Jalabert, Professeur à l'université Louis Pasteur, Institut de Physique et Chimie des Matériaux de Strasbourg, France. Andrey Chubukov, William I. and Bianca M. Fine Chair in Theoretical Physics, University of Minnesota, Minneapolis. Satya Majumdar, Directeur de Recherche, Laboratoire de Physique Théorique et Modèles Statistiques, University of Paris XI, France. Matthias Vojta, Chair of Theoretical Solid State Physics, Technische Universität, Dresden, Germany Oleg Starykh, Professor, Department of Physics, University of Utah. Marcus Kollar, Theoretische Physik III, Institut für Physik, Universität Augsburg, Germany. Kwon Park, Professor, Korea Institute for Advanced Study, Seoul. Takao Morinari, Kyoto University, Kyoto, Japan. Adam Durst, Associate Professor, Hofstra University. Krishnendu Sengupta, Professor, Indian Association for the Cultivation of Science, Kolkata, India. Lorenz Bartosch, Assistant Professor, University of Frankfurt. Predrag Nikolic, Associate Professor, George Mason University Ribhu Kaul, Associate Professor, University of Kentucky Markus Müller, Scientist, Paul Scherrer Institute, Switzerland. Lars Fritz, Assistant Professor, University of Utrecht Michael Levin, Associate Professor, University of Chicago Cenke Xu, Associate Professor, University of California, Santa Barbara Sean Hartnoll, Associate Professor, Stanford University Erez Berg, Associate Professor, University of Chicago Liang Fu, Lawrence C. (1944) and Sarah W. Biedenharn Career Development Associate Professor of Physics, Massachusetts Institute of Technology Liza Huijse, Software Engineer at Karius, Inc. Chris Laumann, Assistant Professor, Boston University Matthias Punk, Faculty, LMU Munich Philipp Strack, ZEISS Group Brian Swingle, Assistant Professor, University of Maryland Dmitry Abanin, Professor of Physics, University of Geneva Ling-Yan (Janet) Hung, Professor of Physics, Fudan University, Shanghai Jay Sau, Assistant Professor, University of Maryland Sarang Gopalakrishnan, Postdoctoral Fellow, Caltech Andrea Allais, Cruise Automation, San Francisco Johannes Bauer, SCL Group, London Paul Chesler, Harvard University Andreas Eberlein, Harvard University William Witczak-Krempa, Assistant Professor, University of Montreal Richard Davison, Harvard University Chong Wang, Harvard University Mathias Scheurer, Harvard University. Postdocs

4 Continuous quantum transitions (A) Broken symmetry No broken symmetry

5 Continuous quantum transitions (A) Broken symmetry No broken symmetry Exact solution by Onsager of the D=2 Ising model Onsager Prizes: M.E. Fisher L. Kadanoff A. I. Larkin V. L. Pokrovsky

6 Continuous quantum transitions (A) Broken symmetry No broken symmetry (B) Topological order No topological order (C) Topological order Broken symmetry (D) Broken symmetry A different broken symmetry

7 Continuous quantum transitions Theory with emergent gauge fields: Higgs/ (A) Broken symmetry No broken symmetry confining phases and phase transitions (B) Topological order No topological order (C) Topological order Broken symmetry (D) Broken symmetry A different broken symmetry

8 Continuous quantum transitions (A) Broken symmetry No broken symmetry (B) Topological order No topological order (C) Topological order Broken symmetry (D) Broken symmetry A different broken symmetry

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sha1_base64="/fbgkyqov5unet6bnprtt569cp4=">aaaccxicdzdntgixfiu7+if4n+rstzwyucizgoi7ohuxmiiqmir0yguaop2x7zcqcws3voobf2rc+gbufbsljikapumtk+/em9t7/igzpr3nw8oslc4tr2rxc2vrg5tb9vbojqpjsafoqx7kpk8uccagrpnm0iwkkmdn0pchf9n6ywrssvbc63ee7yd0besxsrrbhxvf40t0owbvbdtxagm2wz5mkybb7htsvfnwttxkycxgzgtmablydivytukepap17hevg9i4akepj0q1xcfs7yrizsihsc6lfusedkkfwsykeobqj7ntjvjqkc7uhdi8ofgmfp9iskduopbnz0d0qp2utefftvase5v2wkquaxb0vqgxc6xdpm0fd5keqvnygeilm3/fdeakodqklzmhff2k/zf148jzwb0q5avnarpztico0bfyurlv0swqotqi6a49ocf0bn1bj9al9tpvzvjpzc76ievte+dqmfy=</latexit> <latexit sha1_base64="/fbgkyqov5unet6bnprtt569cp4=">aaaccxicdzdntgixfiu7+if4n+rstzwyucizgoi7ohuxmiiqmir0yguaop2x7zcqcws3voobf2rc+gbufbsljikapumtk+/em9t7/igzpr3nw8oslc4tr2rxc2vrg5tb9vbojqpjsafoqx7kpk8uccagrpnm0iwkkmdn0pchf9n6ywrssvbc63ee7yd0besxsrrbhxvf40t0owbvbdtxagm2wz5mkybb7htsvfnwttxkycxgzgtmablydivytukepap17hevg9i4akepj0q1xcfs7yrizsihsc6lfusedkkfwsykeobqj7ntjvjqkc7uhdi8ofgmfp9iskduopbnz0d0qp2utefftvase5v2wkquaxb0vqgxc6xdpm0fd5keqvnygeilm3/fdeakodqklzmhff2k/zf148jzwb0q5avnarpztico0bfyurlv0swqotqi6a49ocf0bn1bj9al9tpvzvjpzc76ievte+dqmfy=</latexit> Antiferromagnetism in the Hubbard Model H = X i<j t ij c i c j + U X i n i" 1 2 n i# 1 2 µ X i c i c i t ij! hopping. U! local repulsion, µ! chemical potential Mean-field theory with a spin density wave (SDW) order parameter ~ D E i =( 1) i x+i y c i ~ c i /2 ncreasing SDW SDW LRO Reconstructed Fermi surface h ~ i6=0 (A) Symmetry breaking phase transition h ~ i =0 SDW SRO Large Fermi surface. U/t

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11 Continuous quantum transitions (A) Broken symmetry No broken symmetry (B) Topological order No topological order (C) Topological order Broken symmetry (D) Broken symmetry A different broken symmetry

12 Z2 lattice gauge theory (Wegner, 1971 Onsager Prize, 2015) z H = X z z z z g X i x z z z G i = x x x x Gauss s Law: [H, G i ]=0, G i =1

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sha1_base64="ae72rwzelrxdyzpaqton0nfbasw=">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</latexit> <latexit sha1_base64="fy9nkk0ijhrx9wayzx00b1bhh3y=">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</latexit> <latexit sha1_base64="fy9nkk0ijhrx9wayzx00b1bhh3y=">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</latexit> <latexit sha1_base64="fy9nkk0ijhrx9wayzx00b1bhh3y=">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</latexit> <latexit sha1_base64="fy9nkk0ijhrx9wayzx00b1bhh3y=">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</latexit> Z2 lattice gauge theory (Wegner, 1971 Onsager Prize, 2015) W C = Y C z C Deconfined phase. Perimeter law for Wegner-Wilson loops Confined phase. Area law for Wegner-Wilson loops g H = X z z z z g X i x

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sha1_base64="gj5u8m98bdnbw8mefkx5wlq/dcg=">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</latexit> <latexit sha1_base64="9qas7ap0+pf++uk1b+sxpie8kyw=">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</latexit> <latexit sha1_base64="9qas7ap0+pf++uk1b+sxpie8kyw=">aaaconicdvhbbtqwehxcryrlf3jkxwjbiqcujvfb4k2iekjcgq1gacvmtxkcya5vx7zsb7gy8mn8bm/8dzptvui6kq3rgz+zc8avkcl5lpsrxdeu37h5a+d2cufuvfu7owcppzndwq5trqw25xvziiwcqrdewrmxwnpkwll1ctluz76adukrj35tyn6yprkn4mwjtbh9k2toklvpfexnjyqvmov7yjdvh7l0ih9gs7ty3ed5n5qvliukhjqh2ycnwg38mpovykjlmgnl381o2aka7kay7jut86uqcp/7rbfhg9l9pczqkrqwzinl+x67vnpua4onlqhvum2rniylqppq9gi0rkvhh0vrolrse5jkgichnn8iy7knywl0vaw171qkc8mcm+wz8foa+gsxgm46b4bxc7aegaaktedmywomp/ui1ltrfo/ydip+ygisdw7dvvhysof+ra3gv2qzzjcv5keo03lq/hjq00l0t4cpo7wwwl1cy8k4faiv8hwzjomoxijluhjk/59mi/rlmp8w4+nx223skmfkcxlkcvkchjm3zekmhec0eh29jybxfvw2po0/xd6noy3nefkt4vinyr/p+a==</latexit> <latexit sha1_base64="9qas7ap0+pf++uk1b+sxpie8kyw=">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</latexit> <latexit sha1_base64="9qas7ap0+pf++uk1b+sxpie8kyw=">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</latexit> Z2 lattice gauge theory Deconfined phase. Z 2 flux expelled. Z 2 (toric code) topological order. (B) Topological phase transition Confined phase. Z 2 flux proliferates. No topological order. g H = X z z z z g X i x E. Fradkin and S. H. Shenker, PRD 19, 3682 (1979); N. Read and S. Sachdev, PRL 66, 1773 (1991); X.-G. Wen, PRB 44, 2664 (1991); A.Y. Kitaev, Annals of Physics 303, 2 (2003)

15 <latexit sha1_base64="4baykf+agtlgymbq0qnkwchxmwm=">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</latexit> <latexit sha1_base64="4baykf+agtlgymbq0qnkwchxmwm=">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</latexit> <latexit sha1_base64="4baykf+agtlgymbq0qnkwchxmwm=">aaaelnicfvpbttragc67hrceqc+9+snrwkksbqkcisbeq8kncuzxsjilze6n7cb02sxmwu1tn8gbx4uby9r462m4bxevneakzfz5/vp3dyyzz0o7ztezvvvs5stxz6/z12/cvhv7bv7oe5xmkta+sxkqd4dyuc4e7wumod3njmxjknod4dglyr9ztkviqxinrxkdjdgslgqeawp5861xaegjjoqaqyzjuyj1xnpb8aywaak88qv2fa5l0mywfzeximyzgjf7kmbrrgufhk5agifqn2swgdgndrdenzznwmr0pkxkhxktwi0lz8rifsv6yz6kddit8b/ock3yv40uguwpacmqguktx5stkjsvpd0z79h0qjt34oagtromiwhxsqed/rstpwrzkdtleuxwi+jum50nitfnwbwvpdr0yzgj9beh+zxfyjlklaegy5rkezwwhqmgzzpbqivi2md4mel37cudjuaiuhmypqyjokfmyrudjikp0hbmsp8zxbrl7tj+wfwprgpmj8bsi45fxon5dduvfissxppf4zw0adxhtlgblzcskz32adonnrpqbh9uwek566676oht85xvz23fge82vbvnhdyeu58fa3y2/bltizhje1oackzunutkelbgqrnh1gitk5phcoqjumdmyvzwg6l+30t4yjaawlsat2io0bmzbu6ugivde1kprf711fkf49vldbg+kjjicimwarqfoqedqvv4igcses1hxsbemjoredjoqqwlqijhsilcbps93kbpfemtbd4fszfr3bpuw13ltdastwvl2rb6fml9ap22vrw+tz+3v7r/th82oa2zcc5da+q0f/0gddxhgw==</latexit> <latexit sha1_base64="4baykf+agtlgymbq0qnkwchxmwm=">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</latexit> <latexit sha1_base64="/fbgkyqov5unet6bnprtt569cp4=">aaaccxicdzdntgixfiu7+if4n+rstzwyucizgoi7ohuxmiiqmir0yguaop2x7zcqcws3voobf2rc+gbufbsljikapumtk+/em9t7/igzpr3nw8oslc4tr2rxc2vrg5tb9vbojqpjsafoqx7kpk8uccagrpnm0iwkkmdn0pchf9n6ywrssvbc63ee7yd0besxsrrbhxvf40t0owbvbdtxagm2wz5mkybb7htsvfnwttxkycxgzgtmablydivytukepap17hevg9i4akepj0q1xcfs7yrizsihsc6lfusedkkfwsykeobqj7ntjvjqkc7uhdi8ofgmfp9iskduopbnz0d0qp2utefftvase5v2wkquaxb0vqgxc6xdpm0fd5keqvnygeilm3/fdeakodqklzmhff2k/zf148jzwb0q5avnarpztico0bfyurlv0swqotqi6a49ocf0bn1bj9al9tpvzvjpzc76ievte+dqmfy=</latexit> <latexit sha1_base64="/fbgkyqov5unet6bnprtt569cp4=">aaaccxicdzdntgixfiu7+if4n+rstzwyucizgoi7ohuxmiiqmir0yguaop2x7zcqcws3voobf2rc+gbufbsljikapumtk+/em9t7/igzpr3nw8oslc4tr2rxc2vrg5tb9vbojqpjsafoqx7kpk8uccagrpnm0iwkkmdn0pchf9n6ywrssvbc63ee7yd0besxsrrbhxvf40t0owbvbdtxagm2wz5mkybb7htsvfnwttxkycxgzgtmablydivytukepap17hevg9i4akepj0q1xcfs7yrizsihsc6lfusedkkfwsykeobqj7ntjvjqkc7uhdi8ofgmfp9iskduopbnz0d0qp2utefftvase5v2wkquaxb0vqgxc6xdpm0fd5keqvnygeilm3/fdeakodqklzmhff2k/zf148jzwb0q5avnarpztico0bfyurlv0swqotqi6a49ocf0bn1bj9al9tpvzvjpzc76ievte+dqmfy=</latexit> <latexit sha1_base64="/fbgkyqov5unet6bnprtt569cp4=">aaaccxicdzdntgixfiu7+if4n+rstzwyucizgoi7ohuxmiiqmir0yguaop2x7zcqcws3voobf2rc+gbufbsljikapumtk+/em9t7/igzpr3nw8oslc4tr2rxc2vrg5tb9vbojqpjsafoqx7kpk8uccagrpnm0iwkkmdn0pchf9n6ywrssvbc63ee7yd0besxsrrbhxvf40t0owbvbdtxagm2wz5mkybb7htsvfnwttxkycxgzgtmablydivytukepap17hevg9i4akepj0q1xcfs7yrizsihsc6lfusedkkfwsykeobqj7ntjvjqkc7uhdi8ofgmfp9iskduopbnz0d0qp2utefftvase5v2wkquaxb0vqgxc6xdpm0fd5keqvnygeilm3/fdeakodqklzmhff2k/zf148jzwb0q5avnarpztico0bfyurlv0swqotqi6a49ocf0bn1bj9al9tpvzvjpzc76ievte+dqmfy=</latexit> <latexit sha1_base64="/fbgkyqov5unet6bnprtt569cp4=">aaaccxicdzdntgixfiu7+if4n+rstzwyucizgoi7ohuxmiiqmir0yguaop2x7zcqcws3voobf2rc+gbufbsljikapumtk+/em9t7/igzpr3nw8oslc4tr2rxc2vrg5tb9vbojqpjsafoqx7kpk8uccagrpnm0iwkkmdn0pchf9n6ywrssvbc63ee7yd0besxsrrbhxvf40t0owbvbdtxagm2wz5mkybb7htsvfnwttxkycxgzgtmablydivytukepap17hevg9i4akepj0q1xcfs7yrizsihsc6lfusedkkfwsykeobqj7ntjvjqkc7uhdi8ofgmfp9iskduopbnz0d0qp2utefftvase5v2wkquaxb0vqgxc6xdpm0fd5keqvnygeilm3/fdeakodqklzmhff2k/zf148jzwb0q5avnarpztico0bfyurlv0swqotqi6a49ocf0bn1bj9al9tpvzvjpzc76ievte+dqmfy=</latexit> Antiferromagnetism in the Hubbard Model H = X i<j t ij c i c j + U X i n i" 1 2 n i# 1 2 µ X i c i c i t ij! hopping. U! local repulsion, µ! chemical potential Mean-field theory with a spin density wave (SDW) order parameter ~ D E i =( 1) i x+i y c i ~ c i /2 ncreasing SDW SDW LRO Reconstructed Fermi surface h ~ i6=0 (A) Symmetry breaking phase transition h ~ i =0 SDW SRO Large Fermi surface. U/t

16 ncreasing SDW <latexit sha1_base64="/fbgkyqov5unet6bnprtt569cp4=">aaaccxicdzdntgixfiu7+if4n+rstzwyucizgoi7ohuxmiiqmir0yguaop2x7zcqcws3voobf2rc+gbufbsljikapumtk+/em9t7/igzpr3nw8oslc4tr2rxc2vrg5tb9vbojqpjsafoqx7kpk8uccagrpnm0iwkkmdn0pchf9n6ywrssvbc63ee7yd0besxsrrbhxvf40t0owbvbdtxagm2wz5mkybb7htsvfnwttxkycxgzgtmablydivytukepap17hevg9i4akepj0q1xcfs7yrizsihsc6lfusedkkfwsykeobqj7ntjvjqkc7uhdi8ofgmfp9iskduopbnz0d0qp2utefftvase5v2wkquaxb0vqgxc6xdpm0fd5keqvnygeilm3/fdeakodqklzmhff2k/zf148jzwb0q5avnarpztico0bfyurlv0swqotqi6a49ocf0bn1bj9al9tpvzvjpzc76ievte+dqmfy=</latexit> <latexit sha1_base64="/fbgkyqov5unet6bnprtt569cp4=">aaaccxicdzdntgixfiu7+if4n+rstzwyucizgoi7ohuxmiiqmir0yguaop2x7zcqcws3voobf2rc+gbufbsljikapumtk+/em9t7/igzpr3nw8oslc4tr2rxc2vrg5tb9vbojqpjsafoqx7kpk8uccagrpnm0iwkkmdn0pchf9n6ywrssvbc63ee7yd0besxsrrbhxvf40t0owbvbdtxagm2wz5mkybb7htsvfnwttxkycxgzgtmablydivytukepap17hevg9i4akepj0q1xcfs7yrizsihsc6lfusedkkfwsykeobqj7ntjvjqkc7uhdi8ofgmfp9iskduopbnz0d0qp2utefftvase5v2wkquaxb0vqgxc6xdpm0fd5keqvnygeilm3/fdeakodqklzmhff2k/zf148jzwb0q5avnarpztico0bfyurlv0swqotqi6a49ocf0bn1bj9al9tpvzvjpzc76ievte+dqmfy=</latexit> <latexit sha1_base64="/fbgkyqov5unet6bnprtt569cp4=">aaaccxicdzdntgixfiu7+if4n+rstzwyucizgoi7ohuxmiiqmir0yguaop2x7zcqcws3voobf2rc+gbufbsljikapumtk+/em9t7/igzpr3nw8oslc4tr2rxc2vrg5tb9vbojqpjsafoqx7kpk8uccagrpnm0iwkkmdn0pchf9n6ywrssvbc63ee7yd0besxsrrbhxvf40t0owbvbdtxagm2wz5mkybb7htsvfnwttxkycxgzgtmablydivytukepap17hevg9i4akepj0q1xcfs7yrizsihsc6lfusedkkfwsykeobqj7ntjvjqkc7uhdi8ofgmfp9iskduopbnz0d0qp2utefftvase5v2wkquaxb0vqgxc6xdpm0fd5keqvnygeilm3/fdeakodqklzmhff2k/zf148jzwb0q5avnarpztico0bfyurlv0swqotqi6a49ocf0bn1bj9al9tpvzvjpzc76ievte+dqmfy=</latexit> <latexit sha1_base64="/fbgkyqov5unet6bnprtt569cp4=">aaaccxicdzdntgixfiu7+if4n+rstzwyucizgoi7ohuxmiiqmir0yguaop2x7zcqcws3voobf2rc+gbufbsljikapumtk+/em9t7/igzpr3nw8oslc4tr2rxc2vrg5tb9vbojqpjsafoqx7kpk8uccagrpnm0iwkkmdn0pchf9n6ywrssvbc63ee7yd0besxsrrbhxvf40t0owbvbdtxagm2wz5mkybb7htsvfnwttxkycxgzgtmablydivytukepap17hevg9i4akepj0q1xcfs7yrizsihsc6lfusedkkfwsykeobqj7ntjvjqkc7uhdi8ofgmfp9iskduopbnz0d0qp2utefftvase5v2wkquaxb0vqgxc6xdpm0fd5keqvnygeilm3/fdeakodqklzmhff2k/zf148jzwb0q5avnarpztico0bfyurlv0swqotqi6a49ocf0bn1bj9al9tpvzvjpzc76ievte+dqmfy=</latexit> SDW SRO Z2 or U(1) topological order. Reconstructed Fermi surface. ncreasing SDW h ~ i =0 (B) Topological phase transition g SDW LRO Reconstructed Fermi surface h ~ i6=0 U/t (A) Symmetry breaking phase transition SDW SRO Large Fermi surface. h ~ i =0

17 ncreasing SDW (B) Topological phase transition SDW SRO Reconstructed Fermi surface with non-luttinger volume. Z2 vortices or hedgehogs expelled. Z2 or U(1) topological order. SDW SRO Large Fermi surface with Luttinger volume. No topological order g Metallic states with non-luttinger volume Fermi surfaces must have topological order T. Senthil, M. Vojta, and S. Sachdev, PRB 69, (2004) S. Sachdev, M. A. Metlitski, Y. Qi, and C. Xu, PRB 80, (2009)

18 ncreasing SDW SDW SRO Reconstructed Fermi surface with non-luttinger volume. Z2 vortices or hedgehogs expelled. Z2 or U(1) topological order. (B) Topological phase transition; phases of a a theory with an emergent SU(2) gauge field. SDW SRO Large Fermi surface with Luttinger volume. No topological order g Metallic states with non-luttinger volume Fermi surfaces must have topological order T. Senthil, M. Vojta, and S. Sachdev, PRB 69, (2004) S. Sachdev, M. A. Metlitski, Y. Qi, and C. Xu, PRB 80, (2009)

19 Change of carrier density at the pseudogap critical point of a cuprate superconductor S. Badoux 1, W. Tabis 2,3, F. Laliberté 2, G. Grissonnanche 1, B. Vignolle 2, D. Vignolles 2, J. Béard 2, D. A. Bonn 4,5, W. N. Hardy 4,5, R. Liang 4,5, N. Doiron-Leyraud 1, Louis Taillefer 1,5 & Cyril Proust 2,5 210 NATURE VOL MARCH 2016 YBa 2 Cu 3 O 6+x

20 ncreasing SDW SDW SRO Reconstructed Fermi surface with non-luttinger volume. Z2 vortices or hedgehogs expelled. Z2 or U(1) topological order. (B) Topological phase transition; phases of a a theory with an emergent SU(2) gauge field. SDW SRO Large Fermi surface with Luttinger volume. No topological order g Can model the doping dependence of the Hall effect in the hole-doped cuprates S. Chatterjee, A. Eberlein, and S. Sachdev, PRB 96, (2017)

21 ncreasing SDW SDW SRO Reconstructed Fermi surface with non-luttinger volume. Z2 vortices or hedgehogs expelled. Z2 or U(1) topological order. (B) Topological phase transition; phases of a a theory with an emergent SU(2) gauge field. SDW SRO Large Fermi surface with Luttinger volume. No topological order g SU(2) gauge theory fits the real and imaginary parts of the electron Green s function computed by multi-site DMFT F : Mathias Scheurer R : Antoine Georges, Thu 10:24 AM M. S. Scheurer, S. Chatterjee, Wei Wu, M. Ferrero, A. Georges, and S. Sachdev, arxiv:

22 ncreasing SDW <latexit sha1_base64="/fbgkyqov5unet6bnprtt569cp4=">aaaccxicdzdntgixfiu7+if4n+rstzwyucizgoi7ohuxmiiqmir0yguaop2x7zcqcws3voobf2rc+gbufbsljikapumtk+/em9t7/igzpr3nw8oslc4tr2rxc2vrg5tb9vbojqpjsafoqx7kpk8uccagrpnm0iwkkmdn0pchf9n6ywrssvbc63ee7yd0besxsrrbhxvf40t0owbvbdtxagm2wz5mkybb7htsvfnwttxkycxgzgtmablydivytukepap17hevg9i4akepj0q1xcfs7yrizsihsc6lfusedkkfwsykeobqj7ntjvjqkc7uhdi8ofgmfp9iskduopbnz0d0qp2utefftvase5v2wkquaxb0vqgxc6xdpm0fd5keqvnygeilm3/fdeakodqklzmhff2k/zf148jzwb0q5avnarpztico0bfyurlv0swqotqi6a49ocf0bn1bj9al9tpvzvjpzc76ievte+dqmfy=</latexit> <latexit sha1_base64="/fbgkyqov5unet6bnprtt569cp4=">aaaccxicdzdntgixfiu7+if4n+rstzwyucizgoi7ohuxmiiqmir0yguaop2x7zcqcws3voobf2rc+gbufbsljikapumtk+/em9t7/igzpr3nw8oslc4tr2rxc2vrg5tb9vbojqpjsafoqx7kpk8uccagrpnm0iwkkmdn0pchf9n6ywrssvbc63ee7yd0besxsrrbhxvf40t0owbvbdtxagm2wz5mkybb7htsvfnwttxkycxgzgtmablydivytukepap17hevg9i4akepj0q1xcfs7yrizsihsc6lfusedkkfwsykeobqj7ntjvjqkc7uhdi8ofgmfp9iskduopbnz0d0qp2utefftvase5v2wkquaxb0vqgxc6xdpm0fd5keqvnygeilm3/fdeakodqklzmhff2k/zf148jzwb0q5avnarpztico0bfyurlv0swqotqi6a49ocf0bn1bj9al9tpvzvjpzc76ievte+dqmfy=</latexit> <latexit sha1_base64="/fbgkyqov5unet6bnprtt569cp4=">aaaccxicdzdntgixfiu7+if4n+rstzwyucizgoi7ohuxmiiqmir0yguaop2x7zcqcws3voobf2rc+gbufbsljikapumtk+/em9t7/igzpr3nw8oslc4tr2rxc2vrg5tb9vbojqpjsafoqx7kpk8uccagrpnm0iwkkmdn0pchf9n6ywrssvbc63ee7yd0besxsrrbhxvf40t0owbvbdtxagm2wz5mkybb7htsvfnwttxkycxgzgtmablydivytukepap17hevg9i4akepj0q1xcfs7yrizsihsc6lfusedkkfwsykeobqj7ntjvjqkc7uhdi8ofgmfp9iskduopbnz0d0qp2utefftvase5v2wkquaxb0vqgxc6xdpm0fd5keqvnygeilm3/fdeakodqklzmhff2k/zf148jzwb0q5avnarpztico0bfyurlv0swqotqi6a49ocf0bn1bj9al9tpvzvjpzc76ievte+dqmfy=</latexit> <latexit sha1_base64="/fbgkyqov5unet6bnprtt569cp4=">aaaccxicdzdntgixfiu7+if4n+rstzwyucizgoi7ohuxmiiqmir0yguaop2x7zcqcws3voobf2rc+gbufbsljikapumtk+/em9t7/igzpr3nw8oslc4tr2rxc2vrg5tb9vbojqpjsafoqx7kpk8uccagrpnm0iwkkmdn0pchf9n6ywrssvbc63ee7yd0besxsrrbhxvf40t0owbvbdtxagm2wz5mkybb7htsvfnwttxkycxgzgtmablydivytukepap17hevg9i4akepj0q1xcfs7yrizsihsc6lfusedkkfwsykeobqj7ntjvjqkc7uhdi8ofgmfp9iskduopbnz0d0qp2utefftvase5v2wkquaxb0vqgxc6xdpm0fd5keqvnygeilm3/fdeakodqklzmhff2k/zf148jzwb0q5avnarpztico0bfyurlv0swqotqi6a49ocf0bn1bj9al9tpvzvjpzc76ievte+dqmfy=</latexit> SDW SRO Z2 or U(1) topological order. Reconstructed Fermi surface. ncreasing SDW h ~ i =0 (B) Topological phase transition g SDW LRO Reconstructed Fermi surface h ~ i6=0 U/t (A) Symmetry breaking phase transition SDW SRO Large Fermi surface. h ~ i =0

23 ncreasing SDW <latexit sha1_base64="/fbgkyqov5unet6bnprtt569cp4=">aaaccxicdzdntgixfiu7+if4n+rstzwyucizgoi7ohuxmiiqmir0yguaop2x7zcqcws3voobf2rc+gbufbsljikapumtk+/em9t7/igzpr3nw8oslc4tr2rxc2vrg5tb9vbojqpjsafoqx7kpk8uccagrpnm0iwkkmdn0pchf9n6ywrssvbc63ee7yd0besxsrrbhxvf40t0owbvbdtxagm2wz5mkybb7htsvfnwttxkycxgzgtmablydivytukepap17hevg9i4akepj0q1xcfs7yrizsihsc6lfusedkkfwsykeobqj7ntjvjqkc7uhdi8ofgmfp9iskduopbnz0d0qp2utefftvase5v2wkquaxb0vqgxc6xdpm0fd5keqvnygeilm3/fdeakodqklzmhff2k/zf148jzwb0q5avnarpztico0bfyurlv0swqotqi6a49ocf0bn1bj9al9tpvzvjpzc76ievte+dqmfy=</latexit> <latexit sha1_base64="/fbgkyqov5unet6bnprtt569cp4=">aaaccxicdzdntgixfiu7+if4n+rstzwyucizgoi7ohuxmiiqmir0yguaop2x7zcqcws3voobf2rc+gbufbsljikapumtk+/em9t7/igzpr3nw8oslc4tr2rxc2vrg5tb9vbojqpjsafoqx7kpk8uccagrpnm0iwkkmdn0pchf9n6ywrssvbc63ee7yd0besxsrrbhxvf40t0owbvbdtxagm2wz5mkybb7htsvfnwttxkycxgzgtmablydivytukepap17hevg9i4akepj0q1xcfs7yrizsihsc6lfusedkkfwsykeobqj7ntjvjqkc7uhdi8ofgmfp9iskduopbnz0d0qp2utefftvase5v2wkquaxb0vqgxc6xdpm0fd5keqvnygeilm3/fdeakodqklzmhff2k/zf148jzwb0q5avnarpztico0bfyurlv0swqotqi6a49ocf0bn1bj9al9tpvzvjpzc76ievte+dqmfy=</latexit> <latexit sha1_base64="/fbgkyqov5unet6bnprtt569cp4=">aaaccxicdzdntgixfiu7+if4n+rstzwyucizgoi7ohuxmiiqmir0yguaop2x7zcqcws3voobf2rc+gbufbsljikapumtk+/em9t7/igzpr3nw8oslc4tr2rxc2vrg5tb9vbojqpjsafoqx7kpk8uccagrpnm0iwkkmdn0pchf9n6ywrssvbc63ee7yd0besxsrrbhxvf40t0owbvbdtxagm2wz5mkybb7htsvfnwttxkycxgzgtmablydivytukepap17hevg9i4akepj0q1xcfs7yrizsihsc6lfusedkkfwsykeobqj7ntjvjqkc7uhdi8ofgmfp9iskduopbnz0d0qp2utefftvase5v2wkquaxb0vqgxc6xdpm0fd5keqvnygeilm3/fdeakodqklzmhff2k/zf148jzwb0q5avnarpztico0bfyurlv0swqotqi6a49ocf0bn1bj9al9tpvzvjpzc76ievte+dqmfy=</latexit> <latexit sha1_base64="/fbgkyqov5unet6bnprtt569cp4=">aaaccxicdzdntgixfiu7+if4n+rstzwyucizgoi7ohuxmiiqmir0yguaop2x7zcqcws3voobf2rc+gbufbsljikapumtk+/em9t7/igzpr3nw8oslc4tr2rxc2vrg5tb9vbojqpjsafoqx7kpk8uccagrpnm0iwkkmdn0pchf9n6ywrssvbc63ee7yd0besxsrrbhxvf40t0owbvbdtxagm2wz5mkybb7htsvfnwttxkycxgzgtmablydivytukepap17hevg9i4akepj0q1xcfs7yrizsihsc6lfusedkkfwsykeobqj7ntjvjqkc7uhdi8ofgmfp9iskduopbnz0d0qp2utefftvase5v2wkquaxb0vqgxc6xdpm0fd5keqvnygeilm3/fdeakodqklzmhff2k/zf148jzwb0q5avnarpztico0bfyurlv0swqotqi6a49ocf0bn1bj9al9tpvzvjpzc76ievte+dqmfy=</latexit> SDW SRO Z2 or U(1) topological order. Reconstructed Fermi surface. (C) Symmetry breaking and topological phase transition ncreasing SDW h ~ i =0 (B) Topological phase transition g SDW LRO Reconstructed Fermi surface h ~ i6=0 U/t (A) Symmetry breaking phase transition SDW SRO Large Fermi surface. h ~ i =0

24 Continuous quantum transitions (A) Broken symmetry No broken symmetry (B) Topological order No topological order (C) Topological order Broken symmetry (D) Broken symmetry A different broken symmetry

25 <latexit sha1_base64="/e7/uuccr029zztltajiuzvtq60=">aaacunicdzjna9taeizx7leipi3bhnmzapf2kjqvq92kuajpitn0keldbcxhvquxvmlqjxzxduzyv7eucsgfyawhzo0o0jzmyogdd2b242hjqgpjkb30wg8epnr8zgxvx1t/+myj/fzfd5oxmuoq5zlxjzezkixcorvw4kmhkwwxxop47poifvwdtrg5+mznbuyzs5wycm6ss8bt0zdgvkiko7ko5/4bk415y+alo4dzyacqm36w6rxucabebrxgwkdmp2g+hqhfhr1uwcfyrpajaleut+q67rrcpe+crh+isu72hrc7tdeg9h1/g2iplmmhdgz0dwbb43rie0fj9q8wyxmzuxeumtgjgby2qpi2gkuc+2fpshdxyymonfqsqxnvsyzzeo2cbca5dktzwlp/tlqsm2awxa4zy3zq/q0tzp/vrqwd7esvuevpufhbgyalbjvdajakqjtmcuye41ryhhfjjohxhys7l8l9ytjv7facr/3o3n5dy4vsklfklqnib7jhdskrgrjofpircu0r78l73xk/5la15tuzl8lf0vq/atxusve=</latexit> <latexit sha1_base64="/e7/uuccr029zztltajiuzvtq60=">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</latexit> <latexit sha1_base64="/e7/uuccr029zztltajiuzvtq60=">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</latexit> <latexit sha1_base64="/e7/uuccr029zztltajiuzvtq60=">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</latexit> Odd Z2 lattice gauge theory z H = X z z z z g X i x z z z G i = x x x x Gauss s Law with background electric charges: [H, G i ]=0, G i = 1

26 <latexit sha1_base64="/e7/uuccr029zztltajiuzvtq60=">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</latexit> <latexit sha1_base64="/e7/uuccr029zztltajiuzvtq60=">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</latexit> <latexit sha1_base64="/e7/uuccr029zztltajiuzvtq60=">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</latexit> <latexit sha1_base64="/e7/uuccr029zztltajiuzvtq60=">aaacunicdzjna9taeizx7leipi3bhnmzapf2kjqvq92kuajpitn0keldbcxhvquxvmlqjxzxduzyv7eucsgfyawhzo0o0jzmyogdd2b242hjqgpjkb30wg8epnr8zgxvx1t/+myj/fzfd5oxmuoq5zlxjzezkixcorvw4kmhkwwxxop47poifvwdtrg5+mznbuyzs5wycm6ss8bt0zdgvkiko7ko5/4bk415y+alo4dzyacqm36w6rxucabebrxgwkdmp2g+hqhfhr1uwcfyrpajaleut+q67rrcpe+crh+isu72hrc7tdeg9h1/g2iplmmhdgz0dwbb43rie0fj9q8wyxmzuxeumtgjgby2qpi2gkuc+2fpshdxyymonfqsqxnvsyzzeo2cbca5dktzwlp/tlqsm2awxa4zy3zq/q0tzp/vrqwd7esvuevpufhbgyalbjvdajakqjtmcuye41ryhhfjjohxhys7l8l9ytjv7facr/3o3n5dy4vsklfklqnib7jhdskrgrjofpircu0r78l73xk/5la15tuzl8lf0vq/atxusve=</latexit> Odd Z2 lattice gauge theory z H = X z z z z g X i x z z z G i = x x x x Gauss s Law with background electric charges: [H, G i ]=0, G i = 1

27 R. Jalabert and S. Sachdev, PRB 44, 686 (1991) T. Senthil, L. Balents, S. Sachdev, A. Vishwanath, and M. P. A. Fisher, PRB 70, (2004) <latexit sha1_base64="bsxulw6fcis26unuhvhdtjairj4=">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</latexit> <latexit sha1_base64="bsxulw6fcis26unuhvhdtjairj4=">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</latexit> <latexit sha1_base64="bsxulw6fcis26unuhvhdtjairj4=">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</latexit> <latexit sha1_base64="bsxulw6fcis26unuhvhdtjairj4=">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</latexit> <latexit sha1_base64="epk4xci9km0tmq/fsycjfnsll/8=">aaacdnicdvbnswmxfmz67fq16tflscpelnmk1r6eogc9klgv2mxjpq9tada7jfmhlp0hxvwrxjyoepxszx9jwiuo6ebgmhndy5sofvwbqt6dsfgjyanpmvl3bn5hcclbxrnusayy1fgienuduq2cs6gzbgrcpwpoham4irrha//qbptmibwwvrscmlylb3fgjzvcb6srqzvlnie0opruxkni8she8tfcbsjmlx56bvise7jx2swksiyykimyqzsxp1ikaisz0htrnbowxtbobnw67ppubdlvhjmbfberaugp69i21c2vnayd5mn7+njtkk3cspr90uch+j2r01jrxhzzyziajv7tdcs/vhpmwgdbzmwagzdsc1ere9gkefaobnifziiejzqpbv+kwycqymwh2rulff2k/ye1urfs9m9lherrqi0ztibw0tby0t6qoln0hmqiovt0jx7rk3pnpdjpzsvn6jgzyqyih3bepwctbpts</latexit> <latexit sha1_base64="epk4xci9km0tmq/fsycjfnsll/8=">aaacdnicdvbnswmxfmz67fq16tflscpelnmk1r6eogc9klgv2mxjpq9tada7jfmhlp0hxvwrxjyoepxszx9jwiuo6ebgmhndy5sofvwbqt6dsfgjyanpmvl3bn5hcclbxrnusayy1fgienuduq2cs6gzbgrcpwpoham4irrha//qbptmibwwvrscmlylb3fgjzvcb6srqzvlnie0opruxkni8she8tfcbsjmlx56bvise7jx2swksiyykimyqzsxp1ikaisz0htrnbowxtbobnw67ppubdlvhjmbfberaugp69i21c2vnayd5mn7+njtkk3cspr90uch+j2r01jrxhzzyziajv7tdcs/vhpmwgdbzmwagzdsc1ere9gkefaobnifziiejzqpbv+kwycqymwh2rulff2k/ye1urfs9m9lherrqi0ztibw0tby0t6qoln0hmqiovt0jx7rk3pnpdjpzsvn6jgzyqyih3bepwctbpts</latexit> <latexit sha1_base64="epk4xci9km0tmq/fsycjfnsll/8=">aaacdnicdvbnswmxfmz67fq16tflscpelnmk1r6eogc9klgv2mxjpq9tada7jfmhlp0hxvwrxjyoepxszx9jwiuo6ebgmhndy5sofvwbqt6dsfgjyanpmvl3bn5hcclbxrnusayy1fgienuduq2cs6gzbgrcpwpoham4irrha//qbptmibwwvrscmlylb3fgjzvcb6srqzvlnie0opruxkni8she8tfcbsjmlx56bvise7jx2swksiyykimyqzsxp1ikaisz0htrnbowxtbobnw67ppubdlvhjmbfberaugp69i21c2vnayd5mn7+njtkk3cspr90uch+j2r01jrxhzzyziajv7tdcs/vhpmwgdbzmwagzdsc1ere9gkefaobnifziiejzqpbv+kwycqymwh2rulff2k/ye1urfs9m9lherrqi0ztibw0tby0t6qoln0hmqiovt0jx7rk3pnpdjpzsvn6jgzyqyih3bepwctbpts</latexit> <latexit sha1_base64="epk4xci9km0tmq/fsycjfnsll/8=">aaacdnicdvbnswmxfmz67fq16tflscpelnmk1r6eogc9klgv2mxjpq9tada7jfmhlp0hxvwrxjyoepxszx9jwiuo6ebgmhndy5sofvwbqt6dsfgjyanpmvl3bn5hcclbxrnusayy1fgienuduq2cs6gzbgrcpwpoham4irrha//qbptmibwwvrscmlylb3fgjzvcb6srqzvlnie0opruxkni8she8tfcbsjmlx56bvise7jx2swksiyykimyqzsxp1ikaisz0htrnbowxtbobnw67ppubdlvhjmbfberaugp69i21c2vnayd5mn7+njtkk3cspr90uch+j2r01jrxhzzyziajv7tdcs/vhpmwgdbzmwagzdsc1ere9gkefaobnifziiejzqpbv+kwycqymwh2rulff2k/ye1urfs9m9lherrqi0ztibw0tby0t6qoln0hmqiovt0jx7rk3pnpdjpzsvn6jgzyqyih3bepwctbpts</latexit> Odd Z2 lattice gauge theory Deconfined phase. Z 2 flux expelled. Z 2 (toric code) topological order. <latexit sha1_base64="gj5u8m98bdnbw8mefkx5wlq/dcg=">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</latexit> <latexit sha1_base64="gj5u8m98bdnbw8mefkx5wlq/dcg=">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</latexit> <latexit sha1_base64="gj5u8m98bdnbw8mefkx5wlq/dcg=">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</latexit> <latexit sha1_base64="gj5u8m98bdnbw8mefkx5wlq/dcg=">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</latexit> (C) Symmetry- breaking and topological phase transition Confined phase. Z 2 flux proliferates. No topological order. Electric field lines lead to symmetry breaking and valence bond solid (VBS) order H = X z z z z g X i x G i = 1 g

28 R. Jalabert and S. Sachdev, PRB 44, 686 (1991) T. Senthil, L. Balents, S. Sachdev, A. Vishwanath, and M. P. A. Fisher, PRB 70, (2004) <latexit sha1_base64="bsxulw6fcis26unuhvhdtjairj4=">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</latexit> <latexit sha1_base64="bsxulw6fcis26unuhvhdtjairj4=">aaadhxichvjda9rafj3er7p+bfxrl8gtuefcerbvvpuwwsep6lbfzbjmjjfzysczywzsuot8el/8k774ookdl+k/8wz3c37ihqmxc+ecofeqtjlcujd87vmxll+5em3jeu/gzvu37/q37x5bxrsoi66lnqcpsycfgpettsjpzycvqystdh7yzu/owfih1wu3qgbsskkjxhdmejpuesmkgxzjs6mmqk0lyqbmulyxrdo2ytcmhtmwijff3ajtjskuqjuclapt9g616vgzrwboi0isxpp8kuatwmvgopvnvliyn0vt5k07jbdolutzwhktrq6gobbb26lkc02drlcoqlosaop0tv2zbo6m4dqxidpakvoqgwvjggrqf2ujzixoijhmhsoou93of6bomatfgazazdsimyxuhx+8err6umw1qwuxs0/7a8xidyeoywwlxby2edbokezrzedwdtttf00yzess6vwya8drwlljgyyel4ditywk8tkryiytyixysbn03nkhigq01wapcnsj/sxowglx7rrvdsha32cd+lfzuhb500kjvfu73h31uf7llsbux6gzmji1xgddubholfizm4xjbrahivxssv/djojgl4hexop9g3uag+q+euc2susekh3ynbyreehew++999h75l/zp/if/s+rq7635twjv5t/7qcqvggq</latexit> <latexit sha1_base64="bsxulw6fcis26unuhvhdtjairj4=">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</latexit> <latexit sha1_base64="bsxulw6fcis26unuhvhdtjairj4=">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</latexit> <latexit sha1_base64="epk4xci9km0tmq/fsycjfnsll/8=">aaacdnicdvbnswmxfmz67fq16tflscpelnmk1r6eogc9klgv2mxjpq9tada7jfmhlp0hxvwrxjyoepxszx9jwiuo6ebgmhndy5sofvwbqt6dsfgjyanpmvl3bn5hcclbxrnusayy1fgienuduq2cs6gzbgrcpwpoham4irrha//qbptmibwwvrscmlylb3fgjzvcb6srqzvlnie0opruxkni8she8tfcbsjmlx56bvise7jx2swksiyykimyqzsxp1ikaisz0htrnbowxtbobnw67ppubdlvhjmbfberaugp69i21c2vnayd5mn7+njtkk3cspr90uch+j2r01jrxhzzyziajv7tdcs/vhpmwgdbzmwagzdsc1ere9gkefaobnifziiejzqpbv+kwycqymwh2rulff2k/ye1urfs9m9lherrqi0ztibw0tby0t6qoln0hmqiovt0jx7rk3pnpdjpzsvn6jgzyqyih3bepwctbpts</latexit> <latexit sha1_base64="epk4xci9km0tmq/fsycjfnsll/8=">aaacdnicdvbnswmxfmz67fq16tflscpelnmk1r6eogc9klgv2mxjpq9tada7jfmhlp0hxvwrxjyoepxszx9jwiuo6ebgmhndy5sofvwbqt6dsfgjyanpmvl3bn5hcclbxrnusayy1fgienuduq2cs6gzbgrcpwpoham4irrha//qbptmibwwvrscmlylb3fgjzvcb6srqzvlnie0opruxkni8she8tfcbsjmlx56bvise7jx2swksiyykimyqzsxp1ikaisz0htrnbowxtbobnw67ppubdlvhjmbfberaugp69i21c2vnayd5mn7+njtkk3cspr90uch+j2r01jrxhzzyziajv7tdcs/vhpmwgdbzmwagzdsc1ere9gkefaobnifziiejzqpbv+kwycqymwh2rulff2k/ye1urfs9m9lherrqi0ztibw0tby0t6qoln0hmqiovt0jx7rk3pnpdjpzsvn6jgzyqyih3bepwctbpts</latexit> <latexit sha1_base64="epk4xci9km0tmq/fsycjfnsll/8=">aaacdnicdvbnswmxfmz67fq16tflscpelnmk1r6eogc9klgv2mxjpq9tada7jfmhlp0hxvwrxjyoepxszx9jwiuo6ebgmhndy5sofvwbqt6dsfgjyanpmvl3bn5hcclbxrnusayy1fgienuduq2cs6gzbgrcpwpoham4irrha//qbptmibwwvrscmlylb3fgjzvcb6srqzvlnie0opruxkni8she8tfcbsjmlx56bvise7jx2swksiyykimyqzsxp1ikaisz0htrnbowxtbobnw67ppubdlvhjmbfberaugp69i21c2vnayd5mn7+njtkk3cspr90uch+j2r01jrxhzzyziajv7tdcs/vhpmwgdbzmwagzdsc1ere9gkefaobnifziiejzqpbv+kwycqymwh2rulff2k/ye1urfs9m9lherrqi0ztibw0tby0t6qoln0hmqiovt0jx7rk3pnpdjpzsvn6jgzyqyih3bepwctbpts</latexit> <latexit sha1_base64="epk4xci9km0tmq/fsycjfnsll/8=">aaacdnicdvbnswmxfmz67fq16tflscpelnmk1r6eogc9klgv2mxjpq9tada7jfmhlp0hxvwrxjyoepxszx9jwiuo6ebgmhndy5sofvwbqt6dsfgjyanpmvl3bn5hcclbxrnusayy1fgienuduq2cs6gzbgrcpwpoham4irrha//qbptmibwwvrscmlylb3fgjzvcb6srqzvlnie0opruxkni8she8tfcbsjmlx56bvise7jx2swksiyykimyqzsxp1ikaisz0htrnbowxtbobnw67ppubdlvhjmbfberaugp69i21c2vnayd5mn7+njtkk3cspr90uch+j2r01jrxhzzyziajv7tdcs/vhpmwgdbzmwagzdsc1ere9gkefaobnifziiejzqpbv+kwycqymwh2rulff2k/ye1urfs9m9lherrqi0ztibw0tby0t6qoln0hmqiovt0jx7rk3pnpdjpzsvn6jgzyqyih3bepwctbpts</latexit> Odd Z2 lattice gauge theory Deconfined phase. Z 2 flux expelled. Z 2 (toric code) topological order. <latexit sha1_base64="gj5u8m98bdnbw8mefkx5wlq/dcg=">aaacu3icdvfnb9qwehxcvwlfcxy5wgyriosijgple0evcobyjjzwbfyrx5nsunvsy3aqrqz8stjxb5hstxiugcnr6m28l/cmlzhc+sz7gcw3bt+5e2/rfvlg4apht0zpn311urmcjlxlbu8r5kakbrmvvirty4g1lyst6vzdmd+5aouevl/8yscszqslgsgzr2g++lhw0cb3rrrmz42ejtdr+2axvr+yddd/q7o0wl/38z4pk1giftgod7zppijvdqo1nus0kpzlesq/5wjzqrrs4dnsly3zy6ok3/p5su0b2v1sudqgjdrp36pcjy0dr63glosaxupua4p6cwwhqbaomiylqpra03w0rqv7e0vroodsxdjkwhu7nn8gy7kp4/noe1lr3rvi55i5n80z42cbbqsuasn3dgzj52wbu2wva8hnwjpjt18hutngw3yup2v0d0zgrxortslnizo7orvaf82mnw8ozkeo03lq/opdtsep13t4k7qwfrixk2wyt8ipb1oyyzjewcv4houk9p/npegp0/xzmt56v7ngfnlbxpidkpo35ih8isdkqnh0gm2jzstid3edn8xyajwonpzn5i+ku199jnkl</latexit> <latexit sha1_base64="gj5u8m98bdnbw8mefkx5wlq/dcg=">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</latexit> <latexit sha1_base64="gj5u8m98bdnbw8mefkx5wlq/dcg=">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</latexit> <latexit sha1_base64="gj5u8m98bdnbw8mefkx5wlq/dcg=">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</latexit> (C) Symmetry- breaking and topological phase transition; phases of a theory with an emergent U(1) gauge field Confined phase. Z 2 flux proliferates. No topological order. Electric field lines lead to symmetry breaking and valence bond solid (VBS) order H = X z z z z g X i x G i = 1 g

29 R. Jalabert and S. Sachdev, PRB 44, 686 (1991) N. Read and S. Sachdev, PRL 66, 1773 (1991) <latexit sha1_base64="bsxulw6fcis26unuhvhdtjairj4=">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</latexit> <latexit sha1_base64="bsxulw6fcis26unuhvhdtjairj4=">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</latexit> <latexit sha1_base64="bsxulw6fcis26unuhvhdtjairj4=">aaadhxichvjda9rafj3er7p+bfxrl8gtuefcerbvvpuwwsep6lbfzbjmjjfzysczywzsuot8el/8k774ookdl+k/8wz3c37ihqmxc+ecofeqtjlcujd87vmxll+5em3jeu/gzvu37/q37x5bxrsoi66lnqcpsycfgpettsjpzycvqystdh7yzu/owfih1wu3qgbsskkjxhdmejpuesmkgxzjs6mmqk0lyqbmulyxrdo2ytcmhtmwijff3ajtjskuqjuclapt9g616vgzrwboi0isxpp8kuatwmvgopvnvliyn0vt5k07jbdolutzwhktrq6gobbb26lkc02drlcoqlosaop0tv2zbo6m4dqxidpakvoqgwvjggrqf2ujzixoijhmhsoou93of6bomatfgazazdsimyxuhx+8err6umw1qwuxs0/7a8xidyeoywwlxby2edbokezrzedwdtttf00yzess6vwya8drwlljgyyel4ditywk8tkryiytyixysbn03nkhigq01wapcnsj/sxowglx7rrvdsha32cd+lfzuhb500kjvfu73h31uf7llsbux6gzmji1xgddubholfizm4xjbrahivxssv/djojgl4hexop9g3uag+q+euc2susekh3ynbyreehew++999h75l/zp/if/s+rq7635twjv5t/7qcqvggq</latexit> <latexit sha1_base64="bsxulw6fcis26unuhvhdtjairj4=">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</latexit> Odd Z2 lattice gauge theory Deconfined phase. Z 2 flux expelled. Z 2 (toric code) topological order. <latexit sha1_base64="gj5u8m98bdnbw8mefkx5wlq/dcg=">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</latexit> <latexit sha1_base64="gj5u8m98bdnbw8mefkx5wlq/dcg=">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</latexit> <latexit sha1_base64="gj5u8m98bdnbw8mefkx5wlq/dcg=">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</latexit> <latexit sha1_base64="gj5u8m98bdnbw8mefkx5wlq/dcg=">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</latexit> (C) Symmetry- breaking and topological phase transition; phases of a theory with an emergent U(1) gauge field Confined phase. Z 2 flux proliferates. No topological order. Electric field lines lead to symmetry breaking and valence bond solid (VBS) order Similar phases and transitions in frustrated square lattice antiferromagnets with spin S=1/2 g

30 Continuous quantum transitions (A) Broken symmetry No broken symmetry (B) Topological order No topological order (C) Topological order Broken symmetry (D) Broken symmetry A different broken symmetry

31 N. Read and S. Sachdev, PRL 62, 1694 (1989) T. Senthil, A. Vishwanath, L. Balents, S. Sachdev, and M. P. A. Fisher, Science 303, 1490 (2004) Quantum phases of a S=1/2 square lattice antiferromagnet (D) Symmetry- breaking to symmetry- breaking transition ) or Antiferromagnet with broken spin rotation symmetry <latexit sha1_base64="1i4sfiptwg+rnc0watd6jqguzra=">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</latexit> <latexit sha1_base64="1i4sfiptwg+rnc0watd6jqguzra=">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</latexit> <latexit sha1_base64="1i4sfiptwg+rnc0watd6jqguzra=">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</latexit> <latexit sha1_base64="1i4sfiptwg+rnc0watd6jqguzra=">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</latexit> VBS with broken lattice rotation symmetry <latexit sha1_base64="sa9scrnmahl1abffamidzhyzxtk=">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</latexit> <latexit sha1_base64="sa9scrnmahl1abffamidzhyzxtk=">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</latexit> <latexit sha1_base64="sa9scrnmahl1abffamidzhyzxtk=">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</latexit> <latexit 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32 N. Read and S. Sachdev, PRL 62, 1694 (1989) T. Senthil, A. Vishwanath, L. Balents, S. Sachdev, and M. P. A. Fisher, Science 303, 1490 (2004) Quantum phases of a S=1/2 square lattice antiferromagnet Antiferromagnet with broken spin rotation symmetry <latexit sha1_base64="1i4sfiptwg+rnc0watd6jqguzra=">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</latexit> <latexit sha1_base64="1i4sfiptwg+rnc0watd6jqguzra=">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</latexit> <latexit sha1_base64="1i4sfiptwg+rnc0watd6jqguzra=">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</latexit> <latexit sha1_base64="1i4sfiptwg+rnc0watd6jqguzra=">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</latexit> (D) Symmetry- breaking to symmetry- breaking transition; phases of a theory with an emergent U(1) gauge field ) or VBS with broken lattice rotation symmetry <latexit sha1_base64="sa9scrnmahl1abffamidzhyzxtk=">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</latexit> <latexit sha1_base64="sa9scrnmahl1abffamidzhyzxtk=">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</latexit> <latexit sha1_base64="sa9scrnmahl1abffamidzhyzxtk=">aaacrxicdvbntxsxepxyvbrli9bjlxyruk/rbvi+ibhupyeggisjyoudjfa89sqelusr/kf+jp4cpj7ac1doict4q6gekinzgr2z5zfvjbmsdqpodza1ptm792h+y/hpyxfpubayeuzmyqw0hfhgxitcgziawihrwuvugwejgvnkcfjnz3+addlouxzm0m54t8uufbw91kl9zwn0pc4faaq7cs8othglryt2awlnadrvhfekyixag2mwdcmsa7tdkifo/1e7txru2nrbbdz3adsixlu1g/hg9janj0idtoqou7tlqrff5vlccecu4yjhdsmt11mwclnhiodiwhtqjp2o6lqhuto11j+ndiy+2uoz89clfjpj2hdvzxx4v9llgd3ddil1xibo8szulrrfq6vyacotcfrdykxlt+do7xb9brnw5v1fdnz2uof9kifc45vmvvk5kfxi5/jinr7ftjqnvuz83kzvh0wcmidfybr5smkyq/bjn3jewksqn+sg/cf/g1/bxxafpdyvtgutzmfyqolhj+bztcw=</latexit> <latexit sha1_base64="sa9scrnmahl1abffamidzhyzxtk=">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</latexit> Phases described by Higgs/confining phases of a U(1) gauge theory, which is deconfined only at criticality g

33 4-spin plaquette singlet state in the Shastry Sutherland compound SrCu 2 (BO 3 ) 2 M. E. Zayed 1,2,3 *, Ch. Rüegg 2,4,5,J.Larrea J. 1,6, A. M. Läuchli 7, C. Panagopoulos 8,9, S. S. Saxena 8, M. Ellerby 5,D.F.McMorrow 5, Th. Strässle 2, S. Klotz 10, G. Hamel 10, R. A. Sadykov 11,12,V.Pomjakushin 2, M. Boehm 13, M. Jiménez Ruiz 13, A. Schneidewind 14, E. Pomjakushina 15, M. Stingaciu 15, K. Conder 15 and H. M. Rønnow 1 Nature Physics 13, (2017)

34 4-spin plaquette singlet state in the Shastry Sutherland compound SrCu2(BO3)2 M. E. Zayed1,2,3*, Ch. Rüegg2,4,5, J. Larrea J.1,6, A. M. Läuchli7, C. Panagopoulos8,9, S. S. Saxena8, M. Ellerby5, D. F. McMorrow5, Th. Strässle2, S. Klotz10, G. Hamel10, R. A. Sadykov11,12, V. Pomjakushin2, M. Boehm13, M. Jiménez Ruiz13, A. Schneidewind14, E. Pomjakushina15, M. Stingaciu15, K. Conder15 and H. M. Rønnow1 S Nature Physics 13, (2017) NATURE PHYSICS DOI: /N Energy (K) Tetra gona l Mon oclin ic Order Laboratory for Quantum Magnetism, Institute of Physics, École Polytechnique Fédérale de Lausanne (EPFL), 1015 Lausanne, Switzerland. 2 Laboratory for Pressure (kbar) Neutron Scattering and Imaging, Paul Scherrer Institut, 5232 Villigen PSI, Switzerland. 3 Department of Physics, Carnegie Mellon University in Qatar, C2 P AFM Temperature (K) The study of interacting spin systems is of fundamental is antiferromagnetically (AFM) ordered, albeit with significant importance for modern condensed-matter physics. On frus- quantum fluctuations that are believed to include resonating trated lattices, magnetic exchange interactions cannot be singlet correlations resulting in fractional excitations16. The phase Dynamics simultaneously satisfied, and often give rise to compet- diagram of the Shastry Sutherland model, both with and without 1 ing exotic ground states. The frustrated two-dimensional applied magnetic field, has been intensively studied by numerous Shastry Sutherland lattice2 realized by SrCu2 (BO3 )2 (refs 3,4) theoretical and numerical approaches4. In the presence of magnetic is an important test case for our understanding of quantum field, magnetization plateaus at fractional values of the saturation magnetism. It was constructed to have an exactly solvable magnetization corresponding to Mott insulator phases of dimer 2-spin dimer singlet ground state within a certain range of states, as well as possible superfluid and supersolid phases have been 80parameters and frustration. While the exact dimer extensively studied7, At zero field, the main unsolved issue is the exchange state and the antiferromagnetic order at both ends of the phase existence and nature of an intermediate phase for diagram are well known, the ground state and spin correlations A variety of quantum phases and transitions between them have in the intermediate frustration range have been widely de- been predicted depending on the theoretical technique used: a bated2,4 14. We report here the first experimental identification direct transition from dimer singlet phase to AFM order2,6,7, or an of the conjectured plaquette singlet intermediate phase in intermediate phase with helical order5, columnar dimers11, valence SrCu2 (BO 603 )2. It is observed by inelastic neutron scattering after bond crystal12 or resonating valence bond plaquettes9,104,20. Recent pressure tuning to 21.5 kbar. This gapped singlet state leads results indicate that a plaquette singlet phase is favoured. From BTlong-range antiferromagnetic order above such a phase, which would have an additional Ising-type order to a transition to 40 kbar, consistent with the existence of a deconfined quantum parameter, a subsequent transition to AFM order could provide a critical point. realization of the so far elusive deconfined quantum critical point21. In the field of quantum magnetism, geometrically frustrated The compound strontium copper borate SrCu2 (BO3 )2 is the only lattices generally imply major difficulties in analytical and known realization of the Shastry Sutherland model with S = 1/2 40studies. For very few particular topologies, however, it spins4 and has thus triggered considerable attention in the field numerical has been shown that the ground state, at least, can be calculated of quantum magnetism. The spectrum of SrCu2 (BO3 )2 exhibits exactly as for the Majumdar Ghosh model15 that solves the J1 J2 an almost dispersionless = 3 mev gap, and a bound state of zigzag chain when J1 = 2J2. In two dimensions, the Shastry two triplets (BT) forms at EBT ' 5 mev. The unusual size and Sutherland model2 consisting of an orthogonal dimer network of dispersionless nature of the gap is an effect of the frustration that spin S = 1/2 was developed to be exactly solvable. For an inter- prevents triplets from hopping up to sixth order4. The estimated 0 dimer J20 to intra-dimer J exchange ratio J 0 /J 0.5 the ground exchange parameters in the material J 85 K and = (ref. 4) state is a product of singlets on the strong bond J. Numerical or J 71 K and = (ref. 8) place the compound close calculations have further shown that this remains valid up to LEto an interesting regime 0.7 where correlations may change 0.7 and for small values of three-dimensional (3D) couplings dramatically at a critical point. AFMa quantum J 00 between dimer layers. AtDimer the other end, for A precious means to tune a quantum magnet across Plaquette the system approaches the well-known 2D square lattice, which phase transition is the application of hydrostatic pressure as

35 4-spin plaquette singlet state in the Shastry Sutherland compound SrCu2(BO3)2 M. E. Zayed1,2,3*, Ch. Rüegg2,4,5, J. Larrea J.1,6, A. M. Läuchli7, C. Panagopoulos8,9, S. S. Saxena8, M. Ellerby5, D. F. McMorrow5, Th. Strässle2, S. Klotz10, G. Hamel10, R. A. Sadykov11,12, V. Pomjakushin2, M. Boehm13, M. Jiménez Ruiz13, A. Schneidewind14, E. Pomjakushina15, M. Stingaciu15, K. Conder15 and H. M. Rønnow1 S Nature Physics 13, (2017) NATURE PHYSICS DOI: /N Energy (K) Tetra gona l Mon oclin ic Order C2 P2 Broken spin rotation invariance Laboratory for Quantum Magnetism, Institute of Physics, École Polytechnique Fédérale de Lausanne (EPFL), 1015 Lausanne, Switzerland. 2 Laboratory for Pressure (kbar) Neutron Scattering and Imaging, Paul Scherrer Institut, 5232 Villigen PSI, Switzerland. 3 Department of Physics, Carnegie Mellon University in Qatar, AFM Temperature (K) The study of interacting spin systems is of fundamental is antiferromagnetically (AFM) ordered, albeit with significant importance for modern condensed-matter physics. On frus- quantum fluctuations that are believed to include resonating trated lattices, magnetic exchange interactions cannot be singlet correlations resulting in fractional excitations16. The phase Dynamics simultaneously satisfied, and often give rise to compet- diagram of the Shastry Sutherland model, both with and without 1 ing exotic ground states. The frustrated two-dimensional applied magnetic field, has been intensively studied by numerous Shastry Sutherland lattice2 realized by SrCu2 (BO3 )2 (refs 3,4) theoretical and numerical approaches4. In the presence of magnetic is an important test case for our understanding of quantum field, magnetization plateaus at fractional values of the saturation magnetism. It was constructed to have an exactly solvable magnetization corresponding to Mott insulator phases of dimer 2-spin dimer singlet ground state within a certain range of states, as well as possible superfluid and supersolid phases have been 80parameters and frustration. While the exact dimer extensively studied7, At zero field, the main unsolved issue is the exchange state and the antiferromagnetic order at both ends of the phase existence and nature of an intermediate phase for diagram are well known, the ground state and spin correlations A variety of quantum phases and transitions between them have in the intermediate frustration range have been widely de- been predicted depending on the theoretical technique used: a bated2,4 14. We report here the first experimental identification direct transition from dimer singlet phase to AFM order2,6,7, or an of the conjectured plaquette singlet intermediate phase in intermediate phase with helical order5, columnar dimers11, valence SrCu2 (BO 603 )2. It is observed by inelastic neutron scattering after bond crystal12 or resonating valence bond plaquettes9,104,20. Recent pressure tuning to 21.5 kbar. This gapped singlet state leads results indicate that a plaquette singlet phase is favoured. From BTlong-range antiferromagnetic order above such a phase, which would have an additional Ising-type order to a transition to 40 kbar, consistent with the existence of a deconfined quantum parameter, a subsequent transition to AFM order could provide a critical point. realization of the so far elusive deconfined quantum critical point21. In the field of quantum magnetism, geometrically frustrated The compound strontium copper borate SrCu2 (BO3 )2 is the only lattices generally imply major difficulties in analytical and known realization of the Shastry Sutherland model with S = 1/2 40studies. For very few particular topologies, however, it spins4 and has thus triggered considerable attention in the field numerical has been shown that the ground state, at least, can be calculated of quantum magnetism. The spectrum of SrCu2 (BO3 )2 exhibits exactly as for the Majumdar Ghosh model15 that solves the J1 J2 an almost dispersionless = 3 mev gap, and a bound state of zigzag chain when J1 = 2J2. In two dimensions, the Shastry two triplets (BT) forms at EBT ' 5 mev. The unusual size and Sutherland model2 consisting of an orthogonal dimer network of dispersionless nature of the gap is an effect of the frustration that spin S = 1/2 was developed to be exactly solvable. For an inter- prevents triplets from hopping up to sixth order4. The estimated 0 dimer J20 to intra-dimer J exchange ratio J 0 /J 0.5 the ground exchange parameters in the material J 85 K and = (ref. 4) state is a product of singlets on the strong bond J. Numerical or J 71 K and = (ref. 8) place the compound close calculations have further shown that this remains valid up to LEto an interesting regime 0.7 where correlations may change 0.7 and for small values of three-dimensional (3D) couplings dramatically at a critical point. AFMa quantum J 00 between dimer layers. AtDimer the other end, for A precious means to tune a quantum magnet across Plaquette the system approaches the well-known 2D square lattice, which phase transition is the application of hydrostatic pressure as

36 4-spin plaquette singlet state in the Shastry Sutherland compound SrCu2(BO3)2 M. E. Zayed1,2,3*, Ch. Rüegg2,4,5, J. Larrea J.1,6, A. M. Läuchli7, C. Panagopoulos8,9, S. S. Saxena8, M. Ellerby5, D. F. McMorrow5, Th. Strässle2, S. Klotz10, G. Hamel10, R. A. Sadykov11,12, V. Pomjakushin2, M. Boehm13, M. Jiménez Ruiz13, A. Schneidewind14, E. Pomjakushina15, M. Stingaciu15, K. Conder15 and H. M. Rønnow1 S Nature Physics 13, (2017) NATURE PHYSICS DOI: /N Energy (K) Tetra gona l Mon oclin ic Order C2 P2 100 Broken lattice symmetry Laboratory for Quantum Magnetism, Institute of Physics, École Polytechnique Fédérale de Lausanne (EPFL), 1015 Lausanne, Switzerland. 2 Laboratory for Pressure (kbar) Neutron Scattering and Imaging, Paul Scherrer Institut, 5232 Villigen PSI, Switzerland. 3 Department of Physics, Carnegie Mellon University in Qatar, 50 AFM Temperature (K) The study of interacting spin systems is of fundamental is antiferromagnetically (AFM) ordered, albeit with significant importance for modern condensed-matter physics. On frus- quantum fluctuations that are believed to include resonating trated lattices, magnetic exchange interactions cannot be singlet correlations resulting in fractional excitations16. The phase Dynamics simultaneously satisfied, and often give rise to compet- diagram of the Shastry Sutherland model, both with and without 1 ing exotic ground states. The frustrated two-dimensional applied magnetic field, has been intensively studied by numerous Shastry Sutherland lattice2 realized by SrCu2 (BO3 )2 (refs 3,4) theoretical and numerical approaches4. In the presence of magnetic is an important test case for our understanding of quantum field, magnetization plateaus at fractional values of the saturation magnetism. It was constructed to have an exactly solvable magnetization corresponding to Mott insulator phases of dimer 2-spin dimer singlet ground state within a certain range of states, as well as possible superfluid and supersolid phases have been 80parameters and frustration. While the exact dimer extensively studied7, At zero field, the main unsolved issue is the exchange state and the antiferromagnetic order at both ends of the phase existence and nature of an intermediate phase for diagram are well known, the ground state and spin correlations A variety of quantum phases and transitions between them have in the intermediate frustration range have been widely de- been predicted depending on the theoretical technique used: a bated2,4 14. We report here the first experimental identification direct transition from dimer singlet phase to AFM order2,6,7, or an of the conjectured plaquette singlet intermediate phase in intermediate phase with helical order5, columnar dimers11, valence SrCu2 (BO 603 )2. It is observed by inelastic neutron scattering after bond crystal12 or resonating valence bond plaquettes9,104,20. Recent pressure tuning to 21.5 kbar. This gapped singlet state leads results indicate that a plaquette singlet phase is favoured. From BTlong-range antiferromagnetic order above such a phase, which would have an additional Ising-type order to a transition to 40 kbar, consistent with the existence of a deconfined quantum parameter, a subsequent transition to AFM order could provide a critical point. realization of the so far elusive deconfined quantum critical point21. In the field of quantum magnetism, geometrically frustrated The compound strontium copper borate SrCu2 (BO3 )2 is the only lattices generally imply major difficulties in analytical and known realization of the Shastry Sutherland model with S = 1/2 40studies. For very few particular topologies, however, it spins4 and has thus triggered considerable attention in the field numerical has been shown that the ground state, at least, can be calculated of quantum magnetism. The spectrum of SrCu2 (BO3 )2 exhibits exactly as for the Majumdar Ghosh model15 that solves the J1 J2 an almost dispersionless = 3 mev gap, and a bound state of zigzag chain when J1 = 2J2. In two dimensions, the Shastry two triplets (BT) forms at EBT ' 5 mev. The unusual size and Sutherland model2 consisting of an orthogonal dimer network of dispersionless nature of the gap is an effect of the frustration that spin S = 1/2 was developed to be exactly solvable. For an inter- prevents triplets from hopping up to sixth order4. The estimated 0 Chung, J.B. Marston, dimer J20 to intra-dimer J exchange ratio J 0 /J 0.5 the groundc.h. exchange parameters in the material J 85 K and = (ref. 4) state is a product of singlets on the strong bond J. Numerical or J and 71 KS. andsachdev, = (ref. 8) place the compound close calculations have further shown that this remains valid up to LEto an interesting regime 0.7 where correlations may change 64, at (2001) 0.7 and for small values of three-dimensional (3D) couplings PRB dramatically a critical point. AFMa quantum J 00 between dimer layers. AtDimer the other end, for A precious means to tune a quantum magnet across Plaquette the system approaches the well-known 2D square lattice, which phase transition is the application of hydrostatic pressure as

37 4-spin plaquette singlet state in the Shastry Sutherland compound SrCu2(BO3)2 M. E. Zayed1,2,3*, Ch. Rüegg2,4,5, J. Larrea J.1,6, A. M. Läuchli7, C. Panagopoulos8,9, S. S. Saxena8, M. Ellerby5, D. F. McMorrow5, Th. Strässle2, S. Klotz10, G. Hamel10, R. A. Sadykov11,12, V. Pomjakushin2, M. Boehm13, M. Jiménez Ruiz13, A. Schneidewind14, E. Pomjakushina15, M. Stingaciu15, K. Conder15 and H. M. Rønnow1 S Nature Physics 13, (2017) NATURE PHYSICS DOI: /N Energy (K) Tetra gona l Mon oclin ic Order C2 P2 (D) Phases described by U(1) gauge theory Laboratory for Quantum Magnetism, Institute of Physics, École Polytechnique Fédérale de Lausanne (EPFL), 1015 Lausanne, Switzerland. 2 Laboratory for Pressure (kbar) Neutron Scattering and Imaging, Paul Scherrer Institut, 5232 Villigen PSI, Switzerland. 3 Department of Physics, Carnegie Mellon University in Qatar, AFM Temperature (K) The study of interacting spin systems is of fundamental is antiferromagnetically (AFM) ordered, albeit with significant importance for modern condensed-matter physics. On frus- quantum fluctuations that are believed to include resonating trated lattices, magnetic exchange interactions cannot be singlet correlations resulting in fractional excitations16. The phase Dynamics simultaneously satisfied, and often give rise to compet- diagram of the Shastry Sutherland model, both with and without 1 ing exotic ground states. The frustrated two-dimensional applied magnetic field, has been intensively studied by numerous Shastry Sutherland lattice2 realized by SrCu2 (BO3 )2 (refs 3,4) theoretical and numerical approaches4. In the presence of magnetic is an important test case for our understanding of quantum field, magnetization plateaus at fractional values of the saturation magnetism. It was constructed to have an exactly solvable magnetization corresponding to Mott insulator phases of dimer 2-spin dimer singlet ground state within a certain range of states, as well as possible superfluid and supersolid phases have been 80parameters and frustration. While the exact dimer extensively studied7, At zero field, the main unsolved issue is the exchange state and the antiferromagnetic order at both ends of the phase existence and nature of an intermediate phase for diagram are well known, the ground state and spin correlations A variety of quantum phases and transitions between them have in the intermediate frustration range have been widely de- been predicted depending on the theoretical technique used: a bated2,4 14. We report here the first experimental identification direct transition from dimer singlet phase to AFM order2,6,7, or an of the conjectured plaquette singlet intermediate phase in intermediate phase with helical order5, columnar dimers11, valence SrCu2 (BO 603 )2. It is observed by inelastic neutron scattering after bond crystal12 or resonating valence bond plaquettes9,104,20. Recent pressure tuning to 21.5 kbar. This gapped singlet state leads results indicate that a plaquette singlet phase is favoured. From BTlong-range antiferromagnetic order above such a phase, which would have an additional Ising-type order to a transition to 40 kbar, consistent with the existence of a deconfined quantum parameter, a subsequent transition to AFM order could provide a critical point. realization of the so far elusive deconfined quantum critical point21. In the field of quantum magnetism, geometrically frustrated The compound strontium copper borate SrCu2 (BO3 )2 is the only lattices generally imply major difficulties in analytical and known realization of the Shastry Sutherland model with S = 1/2 40studies. For very few particular topologies, however, it spins4 and has thus triggered considerable attention in the field numerical has been shown that the ground state, at least, can be calculated of quantum magnetism. The spectrum of SrCu2 (BO3 )2 exhibits exactly as for the Majumdar Ghosh model15 that solves the J1 J2 an almost dispersionless = 3 mev gap, and a bound state of zigzag chain when J1 = 2J2. In two dimensions, the Shastry two triplets (BT) forms at EBT ' 5 mev. The unusual size and Sutherland model2 consisting of an orthogonal dimer network of dispersionless nature of the gap is an effect of the frustration that spin S = 1/2 was developed to be exactly solvable. For an inter- prevents triplets from hopping up to sixth order4. The estimated 0 dimer J20 to intra-dimer J exchange ratio J 0 /J 0.5 the ground exchange parameters in the material J 85 K and = (ref. 4) Chung, Marston, state is a product of singlets on the strong bond J. Numerical or J 71 K and = C.H (ref. 8) place J.B. the compound close LE calculations have further shown that this remains valid up to to an interesting regime 0.7 wheres. correlations may change and Sachdev, 0.7 and for small values of three-dimensional (3D) couplings dramatically at a critical point. AFM J 00 between dimer layers. AtDimer the other end, for A precious means to tune a quantum magnet across a quantum PRB 64, (2001) Plaquette the system approaches the well-known 2D square lattice, which phase transition is the application of hydrostatic pressure as

38 Strongly-coupled quantum criticality States of quantum matter with: No quasiparticle excitations. Strong interactions are at a universal RG fixed point, and this leads so fastest possible dephasing and local thermalization in a time of order ~/(k B T ). S.Sachdev, Quantum Phase Transitions, 1991 J. Zaanen, Nature 430, 512 (2004)

39 <latexit sha1_base64="hppmw3t939/l1p0wwzaggaghxse=">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</latexit> <latexit sha1_base64="hppmw3t939/l1p0wwzaggaghxse=">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</latexit> <latexit sha1_base64="hppmw3t939/l1p0wwzaggaghxse=">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</latexit> <latexit sha1_base64="hppmw3t939/l1p0wwzaggaghxse=">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</latexit> Strongly-coupled quantum criticality States of quantum matter with: No quasiparticle excitations. Strong interactions are at a universal RG fixed point, and this leads so fastest possible dephasing and local thermalization in a time of order ~/(k B T ). Eigenstate thermalization. S.Sachdev, Quantum Phase Transitions, 1991 J. Zaanen, Nature 430, 512 (2004) Many-body quantum chaos (as measured by out-oftime-order correlations) in a time of order ~/(k B T ). J. Maldacena, S. H. Shenker, and D. Stanford, JHEP 08, 106 (2016)

40 <latexit sha1_base64="hppmw3t939/l1p0wwzaggaghxse=">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</latexit> <latexit sha1_base64="hppmw3t939/l1p0wwzaggaghxse=">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</latexit> <latexit sha1_base64="hppmw3t939/l1p0wwzaggaghxse=">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</latexit> <latexit sha1_base64="hppmw3t939/l1p0wwzaggaghxse=">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</latexit> Strongly-coupled quantum criticality States of quantum matter with: No quasiparticle excitations. Strong interactions are at a universal RG fixed point, and this leads so fastest possible dephasing and local thermalization in a time of order ~/(k B T ). Eigenstate thermalization. S.Sachdev, Quantum Phase Transitions, 1991 J. Zaanen, Nature 430, 512 (2004) Many-body quantum chaos (as measured by out-oftime-order correlations) in a time of order ~/(k B T ). J. Maldacena, S. H. Shenker, and D. Stanford, JHEP 08, 106 (2016) Onsager s Ising criticality, other critical states in one spatial dimension, and quantum impurity models, do not have these properties

41 <latexit sha1_base64="hppmw3t939/l1p0wwzaggaghxse=">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</latexit> <latexit sha1_base64="hppmw3t939/l1p0wwzaggaghxse=">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</latexit> <latexit sha1_base64="hppmw3t939/l1p0wwzaggaghxse=">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</latexit> <latexit sha1_base64="hppmw3t939/l1p0wwzaggaghxse=">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</latexit> Strongly-coupled quantum criticality States of quantum matter with: No quasiparticle excitations. Strong interactions are at a universal RG fixed point, and this leads so fastest possible dephasing and local thermalization in a time of order ~/(k B T ). Eigenstate thermalization. S.Sachdev, Quantum Phase Transitions, 1991 J. Zaanen, Nature 430, 512 (2004) Many-body quantum chaos (as measured by out-oftime-order correlations) in a time of order ~/(k B T ). J. Maldacena, S. H. Shenker, and D. Stanford, JHEP 08, 106 (2016) `Solvable models with these properties: the SYK models

42 H = The Sachdev-Ye-Kitaev (SYK) model 1 (2N) 3/2 NX i,j,k,`=1 U ij;k` f i f j f k f` µ X i f i f i X Pick a set of random positions S. Sachdev and J. Ye, Phys. Rev. Lett. 70, 3339 (1993); A. Kitaev (2015)

43 The SYK model H = 1 (2N) 3/2 NX i,j,k,`=1 U ij;k` f i f j f k f` µ X i f i f i X Place electrons randomly on some sites S. Sachdev and J. Ye, Phys. Rev. Lett. 70, 3339 (1993); A. Kitaev (2015)

44 The SYK model H = 1 (2N) 3/2 NX i,j,k,`=1 U ij;k` f i f j f k f` µ X i f i f i X Entangle electrons pairwise randomly S. Sachdev and J. Ye, Phys. Rev. Lett. 70, 3339 (1993); A. Kitaev (2015)

45 The SYK model H = 1 (2N) 3/2 NX i,j,k,`=1 U ij;k` f i f j f k f` µ X i f i f i X Entangle electrons pairwise randomly S. Sachdev and J. Ye, Phys. Rev. Lett. 70, 3339 (1993); A. Kitaev (2015)

46 The SYK model H = 1 (2N) 3/2 NX i,j,k,`=1 U ij;k` f i f j f k f` µ X i f i f i X Entangle electrons pairwise randomly S. Sachdev and J. Ye, Phys. Rev. Lett. 70, 3339 (1993); A. Kitaev (2015)

47 The SYK model H = 1 (2N) 3/2 NX i,j,k,`=1 U ij;k` f i f j f k f` µ X i f i f i X Entangle electrons pairwise randomly S. Sachdev and J. Ye, Phys. Rev. Lett. 70, 3339 (1993); A. Kitaev (2015)

48 The SYK model H = 1 (2N) 3/2 NX i,j,k,`=1 U ij;k` f i f j f k f` µ X i f i f i X Entangle electrons pairwise randomly S. Sachdev and J. Ye, Phys. Rev. Lett. 70, 3339 (1993); A. Kitaev (2015)

49 The SYK model H = 1 (2N) 3/2 NX i,j,k,`=1 U ij;k` f i f j f k f` µ X i f i f i X Entangle electrons pairwise randomly S. Sachdev and J. Ye, Phys. Rev. Lett. 70, 3339 (1993); A. Kitaev (2015)

50 The SYK model H = 1 (2N) 3/2 NX i,j,k,`=1 U ij;k` f i f j f k f` µ X i f i f i X This describes both a strange metal and a black hole! S. Sachdev and J. Ye, Phys. Rev. Lett. 70, 3339 (1993); A. Kitaev (2015)

51 The SYK model Many-body level spacing 2 N = e N ln 2 Non-quasiparticle excitations with spacing e Ns 0 There are 2 N many body levels with energy E, which do not admit a quasiparticle decomposition. Shown are all values of E for a single cluster of size N = 12. The T! 0 state has an entropy S GP S = Ns 0 with s 0 = G + ln(2) 4 < ln 2 = where G is Catalan s constant, for the half-filled case Q =1/2. GPS: A. Georges, O. Parcollet, and S. Sachdev, PRB 63, (2001) W. Fu and S. Sachdev, PRB 94, (2016)

52 The SYK model Low energy, many-body density of states (E) e Ns 0 sinh( p 2(E E 0 )N ) D. Stanford and E. Witten, A. M. Garica-Garcia, J.J.M. Verbaarschot, D. Bagrets, A. Altland, and A. Kamenev, Low temperature entropy S = Ns 0 + N T +... T = 0 fermion Green s function G( ) 1/2 at large. (Fermi liquids with quasiparticles have G( ) 1/ ) S. Sachdev and J. Ye, Phys. Rev. Lett. 70, 3339 (1993) T>0Green sfunction has conformal invariance G (T/sin( k B T /~)) 1/2 A. Georges and O. Parcollet PRB 59, 5341 (1999) The last property indicates eq ~/(k B T ), and this has been found in a recent numerical study. A. Eberlein, V. Kasper, S. Sachdev, and J. Steinberg, arxiv:

53 Black holes share the properties of strongly-coupled quantum criticality Black holes have an entropy and a temperature, T H = ~c 3 /(8 GMk B ). Black holes relax to thermal equilibrium in a time ~/(k B T H )=8 GM/c 3. Black holes

54 <latexit sha1_base64="7avqjth1miftx0lnxteyxsh5lc0=">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</latexit> <latexit sha1_base64="7avqjth1miftx0lnxteyxsh5lc0=">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</latexit> <latexit sha1_base64="7avqjth1miftx0lnxteyxsh5lc0=">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</latexit> <latexit sha1_base64="7avqjth1miftx0lnxteyxsh5lc0=">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</latexit> Black holes share the properties of strongly-coupled quantum criticality Black holes have an entropy and a temperature, T H = ~c 3 /(8 GMk B ). Black holes relax to thermal equilibrium in a time ~/(k B T H )=8 GM/c 3. The entropy of black holes is proportional to their surface area. So they can only be equivalent to quantumcritical systems in one lower dimension. Black holes

55 The SYK model Low energy, many-body density of states (E) e Ns 0 sinh( p 2(E E 0 )N ) D. Stanford and E. Witten, A. M. Garica-Garcia, J.J.M. Verbaarschot, D. Bagrets, A. Altland, and A. Kamenev, Low temperature entropy S = Ns 0 + N T +... T = 0 fermion Green s function G( ) 1/2 at large. (Fermi liquids with quasiparticles have G( ) 1/ ) S. Sachdev and J. Ye, Phys. Rev. Lett. 70, 3339 (1993) T>0Green sfunction has conformal invariance G (T/sin( k B T /~)) 1/2 A. Georges and O. Parcollet PRB 59, 5341 (1999) The last property indicates eq ~/(k B T ), and this has been found in a recent numerical study. A. Eberlein, V. Kasper, S. Sachdev, and J. Steinberg, arxiv:

56 Black hole horizon SYK and black holes ~x Black holes with a near-horizon AdS2 geometry (described by quantum gravity in 1+1 spacetime dimensions) match the properties of the 0+1 dimensional SYK model in the previous slide: Ns0 is the Bekenstein-Hawking entropy S. Sachdev, PRL 105, (2010); A. Kitaev (2015); J. Maldacena, D. Stanford, and Zhenbin Yang, arxiv:

57 <latexit sha1_base64="xqslvroe1fvsrch7xvxtidp0mmm=">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</latexit> <latexit sha1_base64="xqslvroe1fvsrch7xvxtidp0mmm=">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</latexit> <latexit sha1_base64="xqslvroe1fvsrch7xvxtidp0mmm=">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</latexit> <latexit sha1_base64="xqslvroe1fvsrch7xvxtidp0mmm=">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</latexit> Black hole horizon SYK and black holes ~x Both the SYK model and the black hole saturate the lower bound on the Lyapunov time to quantum chaos: L = ~/(2 k B T ). A. Kitaev (2015); J. Maldacena and D. Stanford, arxiv: