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Geometry, topology, and response in condensed matter systems by D´anielVarjas A dissertation submitted in partial satisfaction of the requirements for the degree of Doctor of Philosophy in Physics in the Graduate Division of the University of California, Berkeley Committee in charge: Professor Joel E. Moore, Chair Professor Ashvin Vishwanath Professor Sayeef Salahuddin Summer 2016 Geometry, topology, and response in condensed matter systems Copyright 2016 by D´anielVarjas 1 Abstract Geometry, topology, and response in condensed matter systems by D´anielVarjas Doctor of Philosophy in Physics University of California, Berkeley Professor Joel E. Moore, Chair Topological order provides a new paradigm to view phases of matter. Unlike conven- tional symmetry breaking order, these states are not distinguished by different patterns of symmetry breaking, instead by their intricate mathematical structure, topology. By the bulk-boundary correspondence, the nontrivial topology of the bulk results in robust gap- less excitations on symmetry preserving surfaces. We utilize both of these views to study topological phases together with the analysis of their quantized physical responses to per- turbations. First we study the edge excitations of strongly interacting abelian fractional quantum Hall liquids on an infinite strip geometry. We use the infinite density matrix renormalization group method to numerically measure edge exponents in model systems, including subleading orders. Using analytic methods we derive a generalized Luttinger's theorem that relates momenta of edge excitations. Next we consider topological crystalline insulators protected by space group symme- try. After reviewing the general formalism, we present results about the quantization of the magnetoelectric response protected by orientation-reversing space group symmetries. We construct and analyze insulating and superconducting tight-binding models with glide symmetry in three dimensions to illustrate the general result. Following this, we derive constraints on weak indices of three dimensional topological insulators imposed by space group symmetries. We focus on spin-orbit coupled insulators with and without time reversal invariance and consider both symmorphic and nonsymmorphic symmetries. Finally, we calculate the response of metals and generalize the notion of the magneto- electric effect to noninteracting gapless systems. We use semiclassical dynamics to study the magnetopiezoelectric effect, the current response to elastic strain in static external magnetic fields. i To my parents and grandparents. ii Contents Contents ii I Introduction 1 1 Topological orders with and without symmetry 2 1.1 Symmetry and symmetry breaking . 2 1.2 Topological order . 4 1.2.1 Intrinsic topological order . 4 1.2.2 Symmetry protected topological order . 5 1.3 Topological phases of noninteracting fermions . 7 1.3.1 Quantum anomalous Hall effect and Chern insulators . 7 1.3.2 On-site symmetries and the ten-fold way . 9 1.3.3 Quantum spin Hall effect . 12 1.3.4 Three dimensional topological insulators . 13 1.3.5 Topological crystalline insulators . 13 1.3.6 Weyl semimetals . 14 1.4 Fractional Quantum Hall effect . 15 1.4.1 Bulk theory . 15 1.4.2 Edge theory . 17 1.5 Outlook . 18 2 Overview of dissertation 19 II Edge states of 2D quantum liquids 21 3 Edge excitations of abelian FQH states 22 3.1 Introduction . 22 3.2 Model and Methodology . 24 3.2.1 iDMRG and Finite Entanglement Scaling . 26 3.2.2 Obtaining the edge-exponents . 26 iii 3.3 Edge universality at ν = 1=3; 2=5 and 2=3. 28 3.3.1 The ν = 1=3 edge . 28 3.3.2 Renormalization of edge exponents for thin strips . 29 3.3.3 The ν = 2=5 edge . 31 3.3.4 The ν = 2=3 edge . 32 3.4 Edge states in the 1D picture . 33 3.4.1 A Generalized Luttinger Theorem for Hall Droplets . 33 3.4.2 Adiabatic continuity between Abelian edges . 35 3.5 Conclusion and future directions . 36 3.6 Acknowledgments . 37 III Space group symmetry and topology 38 4 Geometry in band structures of solids 39 4.1 Space group symmetries . 39 4.2 Bloch's theorem . 40 4.2.1 Bloch's theorem and the Brillouin zone . 40 4.2.2 Conventions for Bloch functions . 41 4.3 Electromagnetic response of free fermions . 42 4.4 Differential geometry of band structures . 43 4.4.1 Overview of coordinate-free formalism . 43 4.4.2 Berry curvature . 44 4.4.3 Chern and Chern-Simons forms . 46 4.5 BdG formalism . 48 4.6 Transformation properties . 49 4.6.1 Global symmetries . 49 4.6.2 Space group symmetries . 51 4.6.3 Magnetic space group symmetries . 53 5 θ-terms and topological response with nonsymmorphic symmetries 55 5.1 Introduction . 55 5.2 Magnetoelectric Coupling and Z2 Invariant In Mirror and Glide Symmetric Insulators . 57 5.2.1 Proof of quantization of θ ........................ 58 5.2.2 Mirror symmetry . 60 5.2.3 Glide symmetry . 61 5.3 Topological Crystalline Superconductors in Class D . 63 5.3.1 Bulk invariant and surface thermal Hall conductance . 64 5.3.2 Microscopic model with glide plane in 3d . 64 5.3.3 Lower dimensional topological invariants . 65 5.3.4 Surface Dirac model . 66 iv 5.4 Class C superconductor with glide plane in 3d . 67 5.4.1 Bulk invariant and SU(2) axion term . 67 5.4.2 Microscopic model with glide plane . 69 5.4.3 Surface Dirac model . 70 5.5 Conclusions . 71 5.6 Acknowledgements . 71 Appendix ....................................... 72 5.A Evaluation of Chern-Simons 3-form and second Chern-form . 72 5.B Relation to earlier definition of Z2 index in class A . 75 6 Space group constraints on weak indices in topological insulators 77 6.1 Introduction . 77 6.2 Chern number and Hall conductivity (class A) . 78 6.2.1 Hall conductivity of a 3D insulator . 78 6.2.2 Screw symmetry enforced constraints . 80 6.3 Weak TI indices (class AII) . 82 6.3.1 Bravais lattice . 82 6.3.2 Nonsymorphic symmetries . 83 6.4 Conclusion . 85 6.5 Acknowledgements . 85 Appendix ....................................... 86 6.A Generalization to topological superconductors . 86 6.B No Constraints on Strong TI's . 87 7 Zero and one dimensional invariants in TCI's 89 7.1 Topological protection with restricted equivalence . 89 7.2 Space group representations and zero dimensional invariants . 90 7.2.1 Representation theory of space groups . 90 7.2.2 Classification at fixed filling . 91 7.2.3 Representation valued invariants on simple lattices . 92 7.2.4 Construction of atomic insulators . 93 7.2.5 Invariants with stable equivalence on continuous space . 94 7.3 Wilson loop invariants . 97 7.3.1 Transformation properties of Wilson loops . 97 7.3.2 TCI's from Wilson loop invariants . 99 7.3.3 Derivation of the Chern number formula with Cn symmetry . 99 IV Topological response in metals 101 8 Orbital magnetopiezoelectric effect in metals 102 8.1 Introduction . 102 v 8.2 Methodology . 104 8.3 Experimental feasibility . 108 8.4 Conclusion . 110 8.5 Acknowledgements . 110 Appendix ....................................... 111 i 8.A Vanishing of jc .................................. 111 Bibliography 112 vi Acknowledgments First and foremost, I am grateful to my parents and grandparents for always encouraging my scientific curiosity, their love and continued support throughout all these years of academic pursuit. I dedicate this dissertation to you. I am grateful to those who guided me early on as a physicist: to Istv´anKisp´alfor sowing the first seeds of my love for physics in high school and to Karlo Penc and Gergely Zar´and for advising me during my master's program in Budapest. Without them I would not be completing my graduate studies at Berkeley now. I thank my collaborators, Adolfo Grushin, Roni Ilan, Fernando de Juan, Yuan-Ming Lu, Joel Moore and Mike Zaletel. Research is a cooperative endeavor, which they did not only make possible, but provided guidance, support and entertainment along the way. I also thank my fellow condensed matter theorists, Philipp Dumitrescu, Takahiro Morimoto, Sid Parameswaran, Adrian Po, Andrew Potter, Aaron Szasz and Norman Yao among many others for the enlightening discussions about physics and many other subjects. I am grateful to my friends from Hungary and Berkeley for their support, encouragement, all the great times we had and the wonderful memories we made. I would also like to thank Ashvin Vishwanath and Sayeef Salahuddin for serving on my dissertation committee, and James Analytis for serving on my qualifying exam committee. Finally, I am deeply grateful to my Ph.D. advisor, Joel Moore for his mentorship through- out my graduate studies, his deep insights that guided my research, his unwavering confi- dence in my abilities, and his efforts to secure financial support. My discussions with him profoundly changed the way I view physics and it was a great honor to be his student. 1 Part I Introduction 2 Chapter 1 Topological orders with and without symmetry 1.1 Symmetry and symmetry breaking The huge success of characterizing phases of matter in the 20th century relied on Lev Lan- dau's insight[1]: phases of matter are distinguished by the symmetry of ordering. The underlying Hamiltonian of our physical world has a large symmetry, for example continuous translation and rotation invariance. Solid state systems, however, develop a crystal structure that only retains discrete translations and rotations. The symmetry of the high temperature phase, such as gas or liquid gets spontaneously broken at lower temperatures in order to find the minimum energy configuration. This is a fight between energy and entropy, the system tries to minimize its free energy F = E − TS. At high temperatures a disordered phase with high entropy is favorable, that, on average, is highly symmetric. As T is lowered, energy minimization becomes dominant and a highly ordered phase with low entropy is selected.

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