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Short-Range Entangled Phases of Fermions

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Short-Range Entangled Phases of Fermions Short-Range Entangled Phases of Fermions Thesis by Alexander M. Turzillo In Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy in Physics CALIFORNIA INSTITUTE OF TECHNOLOGY Pasadena, California 2019 Defended May 16, 2019 ii © 2019 Alexander M. Turzillo ORCID: 0000-0003-4293-4293 All rights reserved iii ACKNOWLEDGEMENTS First and foremost, I owe a great debt of gratitude to my advisor Anton Kapustin for his steadfast support and patient guidance over the past few years. I am grateful for the diverse and interesting ideas and insights our conversations exposed me to and for the intuitions they instilled in me. I consider myself enormously lucky to have had the chance to be his student. I owe appreciation to the United States Department of Energy, which supported the work represented here through the grant de-sc0011632, and likewise the Simons Investigator Award. I have been exceedingly lucky to have traveled and learned an enormous amount from researchers outside of Caltech; this was made possible by the generous hospitality and financial support of institutions such as TASI and the University of Colorado Boulder, the Yukawa Institute for Theoretical Physics, the Perimeter Institute, and Fudan University as well as support from Caltech’s Grad Office and Groce Conference Travel Fund. I am grateful for the support I received from my committee members Xie Chen, Matilde Marcolli, and Olexei Motrunich. Many thanks to John Barrett, Yu-An Chen, Po-Shen Hsin, Sujeet Shukla, Ryan Thorngren, Zitao Wang, and Minyoung You for enjoyable collaboration on the work that makes up this thesis (and some that doesn’t). I had the honor of knowing Caltech students, postdocs, and visitors, in addition to those mentioned already, who challenged my ideas and inspired new interests within physics, especially Dave Aasen, Aidan Chatwin-Davies, Matt Heydeman, Nick Hunter-Jones, Murat Koloğlu, Petr Kravchuk, Ingmar Saberi, Burak Şahinoğlu, and Dom Williamson. Finally, I would also like to thank the friends I made along the way and many people who have stuck by me from my life before Caltech. iv ABSTRACT This thesis is an attempt to understand the physics of short-range entangled phases of fermions through several related approaches. The first angle is topological quantum field theory. We discuss the classification of interacting fermionic short-range entan- gled phases by spin cobordism and give an algebraic characterization of unoriented equivariant bosonic topological quantum field theories in one spatial dimension. A second tool is tensor network representation. We develop the formalism of fermionic matrix product states and use it to derive the stacking group law for one dimensional symmetry-enriched fermionic short-range entangled phases. We also study its re- lationship with state sum constructions of topological quantum field theories and develop a state sum construction for pin-minus theories in one spatial dimension. The third approach is topological band theory. We classify free fermionic phases enriched by a unitary symmetry in any dimension and determine the map into the interacting classification. v PUBLISHED CONTENT AND CONTRIBUTIONS Much of this thesis is adapted from articles that originally appeared in other forms. All are available as arXiv preprints; where appropriate, the DOI for the published version is included as a hyperlink from the journal citation. Chapter 1 is based on Kapustin, A., R. Thorngren, A. Turzillo, and Z. Wang (2015). “Fermionic symmetry protected topological phases and cobordisms”. In: JHEP 2015.12. doi: 10.1007/ JHEP12(2015)052. arXiv: 1406.7329. All authors contributed equally. Chapter 2 is based on Kapustin, A. and A. Turzillo (2017). “Equivariant TQFT and Symmetry Protected Topological Phases”. In: JHEP 2017.6. doi: 10 . 1007 / JHEP03(2017 ) 006. arXiv: 1504.01830. Both authors contributed equally. Chapter 3 is based on Kapustin, A., A. Turzillo, and M. You (2017). “Topological field theory and matrix product states”. In: Phys. Rev. B 96 (7), p. 075125. doi: 10.1103/PhysRevB. 96.075125. arXiv: 1607.06766. to which all authors contributed equally, Kapustin, A., A. Turzillo, and M. You (2018). “Spin topological field theory and fermionic matrix product states”. In: Phys. Rev. B 98 (12), p. 125101. doi: 10. 1103/PhysRevB.98.125101. arXiv: 1610.10075. to which all authors contributed equally, and Turzillo,A. and M. You(2019). “Fermionic matrix product states and one-dimensional short-range entangled phases with antiunitary symmetries”. In: Phys. Rev. B 99 (3), p. 035103. doi: 10.1103/PhysRevB.99.035103. arXiv: 1710.00140. to which both authors contributed equally. Chapter 4 is based on Turzillo, A. (2018). “Diagrammatic state sums for two dimensional pin-minus topo- logical quantum field theories”. In: arXiv: 1811.12654. Finally, Chapter 5 is based on vi Chen, Y-A. et al. (2018). “Free and interacting short-range entangled phases of fermions: beyond the ten-fold way”. In: arXiv: 1809.04958. All authors contributed equally. vii TABLE OF CONTENTS Acknowledgements . iii Abstract . iv Published Content and Contributions . v Table of Contents . vii Introduction . 1 Chapter I: Fermionic Symmetry Protected Topological Phases and Cobordisms 5 1.1 Spin and Pin structures . 6 1.2 Working assumptions . 8 1.3 Fermionic SPT phases without any symmetry . 10 1.4 Fermionic SPT phases with time-reversal symmetry . 13 1.5 Fermionic SPT phases with a unitary Z2 symmetry . 17 1.6 Fermionic SPT phases with a general symmetry . 19 1.7 Decorated Domain Walls . 23 1.8 Concluding remarks . 25 Chapter II: Equivariant Topological Quantum Field Theory and Symmetry Protected Topological Phases . 27 2.1 Oriented equivariant TQFT . 30 2.2 Unoriented equivariant D = 0 TQFT . 34 2.3 Unoriented equivariant D = 1 TQFT . 35 Chapter III: Topological Field Theory, Matrix Product States, and the Stacking Law in One Dimension . 40 3.1 Matrix Product States at RG Fixed Points . 43 3.2 Topological Quantum Field Theory . 49 3.3 Equivariant TQFT and Equivariant MPS . 59 3.4 Spin-TQFTs . 70 3.5 Fermionic MPS . 77 3.6 Hamiltonians for fermionic SRE phases . 85 3.7 Equivariant spin-TQFT and equivariant fermionic MPS . 88 3.8 Fermionic MPS . 99 3.9 The fermionic stacking law . 110 3.10 Examples . 114 Chapter IV: Diagrammatic State Sums for 2D Pin-Minus TQFTs . 119 4.1 Pin Geometry in Two Dimensions . 121 4.2 Ribbon Diagrams and Half Twist Algebras . 126 4.3 Real Superalgebras and the Arf-Brown-Kervaire TQFT . 134 Chapter V: Free and Interacting SRE Phases of Fermions: Beyond the Ten- Fold Way . 147 5.1 Free fermionic systems with a unitary symmetry . 150 5.2 Interacting invariants of band Hamiltonians . 155 viii Bibliography . 166 Appendix A: Appendix for Chapter 1 . 173 Appendix B: Appendices for Chapter 2 . 175 B.1 Proof of Proposition 1 . 175 B.2 Proof of Proposition 2 . 178 Appendix C: Appendices for Chapter 3 . 185 C.1 Diagrams for the ground states . 185 C.2 Necessity of supercommutativity . 187 C.3 Appendix: Description of ! in terms of pairs ¹α; βº . 189 C.4 Derivation of the group law for fermionic SRE phases . 189 C.5 Relations between bosonic and fermionic invariants . 192 Appendix D: Appendices for Chapter 5 . 197 D.1 Appendix: Pin groups . 197 D.2 Appendix: Characteristic classes of representations of finite groups . 198 D.3 Appendix: Beta as a charge pumping invariant . 200 1 INTRODUCTION Understanding the highly entangled low energy states of many-body systems is a core problem of modern condensed matter physics. Strong correlations and massive superpositions are defining characteristics of the zoo of physical systems beyond the Landau paradigm, from topological insulators and symmetry protected phases, to topological orders, to quantum spin liquids, and beyond. With experimental realization of these exotic phases of matter has come an interest in theoretical models that capture their universal behaviors as well as proposals for new phases based on arguments in group theory and topology. The projects of classification and characterization require novel techniques to handle the large number of degrees of freedom and lack of local order parameters. In studies of many-body systems, one is typically interested in families (called “phases”) of systems, defined by shared qualitative features. These features are taken to be those of the long distance theories obtained by renormalization group flow or, equivalently, those invariant under under appropriate deformations of the microscopic systems. This thesis will focus on many-body systems that are gapped in the thermodynamic limit; in this case, the appropriate deformations are those that preserve the gap. While many gapped phases – the topological orders – exhibit exotic behavior like topology-dependent ground state degeneracy, anyonic excitations, and long-range entanglement, there are others – the short-range entangled (SRE) phases – that do not yet are nonetheless distinct from the “trivial” phase containing the product state ground state. A pair of gapped systems may be stacked to obtain another gapped system. This operation is commutative, is compatible with the phase equivalence relation, and has the trivial phase as its unit. According to a standard definition, SRE phases (also called “invertible phases”) are those that are invertible with respect to this operation; hence, stacking defines the structure of an abelian group on the classification of SRE phases. It is interesting to study gapped systems that have been enriched by symmetry. In this context, the phase equivalence is modified so that the gap as well as the symmetry is preserved along the deformation. A related concept is that of symmetry protected trivial (SPT) phases: those that become trivial upon forgetting symmetry. In contexts (i.e. dimensions, bosons/fermions) where there exist nontrivial SRE phases without 2 symmetry enrichment (sometimes called invertible topological orders (iTO)), not all symmetry-enriched SRE phases are SPT phases. A 2013 proposal (X. Chen, Gu, Z.-X.
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