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The Chern–Simons Action & Quantum Hall Effect

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The Chern–Simons Action & Quantum Hall Effect Imperial College London Theoretical Physics Group — Department of Physics MSc Quantum Fields & Fundamental Forces The Chern–Simons Action & Quantum Hall Effect: Effective Theory, Anomalies, and Dualities ofa Topological Quantum Fluid Author: Josef Willsher Supervisor: Dr. Matthew M. Roberts Submitted in partial fulfilment of the requirements for the degree of Master of Science of Imperial College London. Word Count: 36,779 September 2020 i Acknowledgements I would like to thank all of my friends for being there for me over the last months; and especially my fellow QFFF friends. I’m lucky to have such a warm community of people who stood up for each other over lockdown, and have been there for me over the summer. Thank you to Noam and Nathan for reading through sections ofthis dissertation, and to many others for giving me a chance to have fun during this pretty tough summer. Thank you to my supervisor, Matt, for giving me a chance to work onsuchan interesting topic, and for your support explaining much of this theory to me. I am grateful to Prof. Knolle for hosting me at TU, Munich for the last month of my work on this dissertation, and everybody else I’ve met here. I’m looking forward for working together during the upcoming years. Most of all I would like to thank my partner Felix, who has helped me more than anyone this year and to whom I owe everything. ii Abstract This dissertation will examine the use of quantum field theory in the paradig- matic setting of the quantum Hall effect. This work aims to provide a comprehen- sive review of the use of non-perturbative field theory methods in the context of the Chern–Simons effective action of this system. Particular emphasis is placed on the role of anomalies in 2+1 dimensions and their responsibility in producing gapless, chiral edge excitations. The canonical quantisation of the theory allows for the full description of the edge modes of Laughlin states, and its spectrum is described using conformal field theory techniques. This is extended by the work of Moore, Read, and Witten to detail the duality between the topological Chern–Simons action and its conformal boundary — a process which allows for the calculation of the bulk wavefunction. The emergence of anyons with frac- tionalised charge and statistics is scrutinised, and particular attention is paid to the contradiction in the literature arising from the flux quantisation condition of the statistical gauge field describing these particles. Bosonisation is discussed throughout: first exactly by examining 1+1 dimensional edge excitations, and then by following a recent field of work which provides a bosonisation duality in 2+1 dimensions that relates particle and vortex excitations. This is finally applied to the half-filled lowest Landau level to motivate the pioneering Dirac composite fermion theory of Son. Throughout this work, attention is paid to the validity of dualities and effective theories away from the zero-magnetic field limit, an often neglected regime which is of the utmost physical relevance in the quantum Hall setting. Contents 1 The Quantum Hall Effect 1 1.1 Introduction ............................... 1 1.2 Integer QHE .............................. 5 1.2.1 Landau Levels ......................... 5 1.2.2 Transport & The IQHE Edge ................. 8 1.3 Fractional QHE ............................. 11 1.3.1 Laughlin Wavefunction ..................... 11 1.3.2 Fractional Statistics ....................... 13 2 The Chern–Simons Effective Action 17 2.1 Chern–Simons Theory of the IQHE . 17 2.1.1 The Chern–Simons Action ................... 17 2.1.2 Gauge Invariance ........................ 18 2.1.3 Emergence of the QHE .................... 21 2.1.4 Chern–Simons Effective Action & Parity Anomaly . 22 2.1.5 Effective Action at 퐵 ≠ 0 .................... 25 2.2 Quantum Hall Boundary Anomaly ................... 28 2.2.1 Relativistic Fermions from Band Theory . 28 2.2.2 Chiral Anomaly ......................... 30 2.2.3 Anomaly Inflow ........................ 33 2.3 Chern–Simons Theory of the FQHE . 36 2.3.1 Statistical Gauge Field ..................... 36 2.3.2 Effective Theory and the FQHE . 38 2.3.3 Flux Quantisation of the Statistical Gauge Field . 39 2.4 Anyons & Topological Order ...................... 40 3 Bosonisation of the Boundary Excitations 45 3.1 1+1 Dimensional Bosonisation ..................... 45 3.1.1 Luttinger Liquid ........................ 45 3.1.2 Schwinger Terms ........................ 48 3.1.3 Boson Duality ......................... 50 3.2 Stone Excitations of Edge ....................... 53 3.3 FQHE Boundary Excitations ...................... 57 3.3.1 FQHE Boundary Anomaly . 57 3.3.2 Hydrodynamic Theory of Edge States . 60 3.3.3 Charge Excitations ....................... 62 iv CONTENTS v 4 Bulk-Boundary Correspondence 65 4.1 Conformal Field Theory ........................ 65 4.1.1 Conformal Group in 푑 Dimensions . 65 4.1.2 Conformal Group in 2 Dimensions . 66 4.1.3 Operator Product Expansion . 68 4.1.4 Conformal Anomaly ...................... 70 4.2 Quantum Hall Correspondence .................... 71 4.2.1 Motivation ........................... 71 4.2.2 Chiral Boson .......................... 73 4.2.3 Vertex Operators & Fusion Rules . 75 4.2.4 Correspondence ......................... 77 4.3 Non-Abelian Quantum Hall Systems . 80 5 Particle-Vortex Duality in the FQHE 85 5.1 Dualities in Quantum Field Theory ................... 85 5.1.1 Photon – Compact Scalar Duality . 86 5.1.2 XY Model ............................ 87 5.1.3 Abelian Higgs Model ...................... 89 5.1.4 XY – Abelian Higgs Duality . 92 5.2 Vortices in the Zhang–Hansson–Kivelson Model . 94 5.3 Composite Fermions .......................... 96 5.3.1 Halperin–Lee–Reed Theory . 96 5.3.2 Dirac Composite Fermion ................... 98 5.4 Composite Fermion Duality . 102 5.4.1 2+1D Bosonisation . 102 5.4.2 Fermionic Particle-Vortex Duality . 105 6 Discussion & Conclusion 109 A Fundamentals 115 B 1+1D Chiral Anomaly 121 C The Wen–Zee Model & Hierarchy States 125 D Diagonalising the Luttinger Hamiltonian 127 E Deriving the Seed Duality 129 Bibliography 133 The Quantum Hall Effect 1 “It is fortunate that solid-state and many-body theorists have so far been spared the plagues of quantum field theory” Mattis and Lieb (1965) 1.1 Introduction In much of theoretical physics, the study of spaces which do not have one temporal and three spatial dimensions amounts to an abstract act of curiosity. Indeed one finds a rich catalogue physical phenomena which only become possible outside of 3+1 dimensional spacetime, but Nature does not allow us to probe these behaviours quite so simply. Far from being abstract, the physics of electrons in a humble semiconductor pro- vides a revolutionary theoretical playground for the behaviour of fermions in lower dimensions: the quantum Hall effect (QHE). In a process well established since the 1980s, the boundary between semiconducting silicon and its insulating oxide allows for the confinement of electrons in a two-dimensional ‘inversion layer’. The application of a strong magnetic field adds further complexity by splitting the spectrum of the two-dimensional electrons into Landau levels separated by large energy gaps. The well-known classical result of passing a current through such a system isto produce a ‘Hall voltage’ perpendicular to the current. However at high magnetic field, the behaviour of this system fundamentally changes as quantum mechanics dominate. Experimentally this was first noticed when the measured Hall conductivity of the system showed noticeable plateaus at precise multiples of a quantum of conductivity, as shown in Fig. 1.1 (von Klitzing et al., 1980). The plateaus of Hall conductivity are interpreted as occurring at exact filling of 휈 Landau levels, and at these points the longitudinal conductivity drops to zero because there are no partially-filled levels which are able to conduct. The fractional quantum Hall effect was also observed within two years of this initial observation (Tsui et al., 1982); this phase was first identified by a 1 휈 = 3 plateau of the Hall conductivity. Strong interactions between the electrons of this system stabilise the system when a third of the lowest Landau level is filled, which introduces a small gap in energy to the filling of additional states. This gap leads toa plateau in the conductivity for a small range of filling fractions around a third. Nowadays the quantum Hall system is a ubiquitous and all-important tool for metrology as it provides an unparalleled measurement of the fine structure constant, 1 . The Quantum Hall Effect Figure .: Hall resistivity 휌푥푦 plotted as the filling fraction is varied (through exper- imentally varying the magnetic field 퐵). The visible plateaus are the hallmarks of the quantum Hall effect. Reproduced from the Nobel prize lecture by Tsui (1998). and its theoretical study has won two Nobel prizes and spurred countless fields of study in the condensed matter community (von Klitzing, 2017). This dissertation concerns the description of this quantum regime of the Hallef- fect using field theory methods. Quantum field theory is a stunning framework which has been applied to understanding fundamental physics with astounding precision. In the field of condensed matter physics, field theory is equally powerful but perhaps less intuitive since one does not get to use the familiar inventory of fields and symmetries which describe Nature. We will aim to describe the quantum Hall effect with an ef- fective theory of the background electromagnetic (EM) gauge field 퐴 in the sample. The interactions of the electrons and the magnetic field will be encoded in theaction of this effective theory. The beauty of physics in these strongly interacting materials, however, is that there may arise emergent degrees of freedom which one may describe with quantum fields which provide an even larger inventory of possible theories. In the fractional quantum Hall system, the emergent degree of freedom is a dynamical gauge field, denoted 푎, which couples to the applied background EM field 퐴 in a way which gives it fractional charge.
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