DOCSLIB.ORG

The Chern–Simons Action & Quantum Hall Effect

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text

Load more

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.
Recommended publications
  • Effective Quantum Field Theories Thomas Mannel Theoretical Physics I (Particle Physics) University of Siegen, Siegen, Germany

    Effective Quantum Field Theories Thomas Mannel Theoretical Physics I (Particle Physics) University of Siegen, Siegen, Germany

    Generating Functionals Functional Integration Renormalization Introduction to Effective Quantum Field Theories Thomas Mannel Theoretical Physics I (Particle Physics) University of Siegen, Siegen, Germany 2nd Autumn School on High Energy Physics and Quantum Field Theory Yerevan, Armenia, 6-10 October, 2014 T. Mannel, Siegen University Effective Quantum Field Theories: Lecture 1 Generating Functionals Functional Integration Renormalization Overview Lecture 1: Basics of Quantum Field Theory Generating Functionals Functional Integration Perturbation Theory Renormalization Lecture 2: Effective Field Thoeries Effective Actions Effective Lagrangians Identifying relevant degrees of freedom Renormalization and Renormalization Group T. Mannel, Siegen University Effective Quantum Field Theories: Lecture 1 Generating Functionals Functional Integration Renormalization Lecture 3: Examples @ work From Standard Model to Fermi Theory From QCD to Heavy Quark Effective Theory From QCD to Chiral Perturbation Theory From New Physics to the Standard Model Lecture 4: Limitations: When Effective Field Theories become ineffective Dispersion theory and effective field theory Bound Systems of Quarks and anomalous thresholds When quarks are needed in QCD É. T. Mannel, Siegen University Effective Quantum Field Theories: Lecture 1 Generating Functionals Functional Integration Renormalization Lecture 1: Basics of Quantum Field Theory Thomas Mannel Theoretische Physik I, Universität Siegen f q f et Yerevan, October 2014 T. Mannel, Siegen University Effective Quantum
  • Bosonization and Mirror Symmetry Arxiv:1608.05077V1 [Hep-Th]

    Bosonization and Mirror Symmetry Arxiv:1608.05077V1 [Hep-Th]

    Bosonization and Mirror Symmetry Shamit Kachru1, Michael Mulligan2,1, Gonzalo Torroba3, and Huajia Wang4 1Stanford Institute for Theoretical Physics, Stanford University, Stanford, CA 94305, USA 2Department of Physics and Astronomy, University of California, Riverside, CA 92511, USA 3Centro At´omico Bariloche and CONICET, R8402AGP Bariloche, ARG 4Department of Physics, University of Illinois, Urbana-Champaign, IL 61801, USA Abstract We study bosonization in 2+1 dimensions using mirror symmetry, a duality that relates pairs of supersymmetric theories. Upon breaking supersymmetry in a controlled way, we dynamically obtain the bosonization duality that equates the theory of a free Dirac fermion to QED3 with a single scalar boson. This duality may be used to demonstrate the bosonization duality relating an O(2)-symmetric Wilson-Fisher fixed point to QED3 with a single Dirac fermion, Peskin-Dasgupta-Halperin duality, and the recently conjectured duality relating the theory of a free Dirac fermion to fermionic QED3 with a single flavor. Chern-Simons and BF couplings for both dynamical and background gauge fields play a central role in our approach. In the course of our study, we describe a \chiral" mirror pair that may be viewed as the minimal supersymmetric generalization of the two bosonization dualities. arXiv:1608.05077v1 [hep-th] 17 Aug 2016 Contents 1 Introduction 1 2 Mirror symmetry and its deformations 4 2.1 Superfields and lagrangians . .5 2.2 = 4 mirror symmetry . .7 N 2.3 Deformations by U(1)A and U(1)R backgrounds . 10 2.4 General mirror duality . 11 3 Chiral mirror symmetry 12 3.1 Chiral theory A .
  • Large-Deviation Principles, Stochastic Effective Actions, Path Entropies

    Large-Deviation Principles, Stochastic Effective Actions, Path Entropies

    Large-deviation principles, stochastic effective actions, path entropies, and the structure and meaning of thermodynamic descriptions Eric Smith Santa Fe Institute, 1399 Hyde Park Road, Santa Fe, NM 87501, USA (Dated: February 18, 2011) The meaning of thermodynamic descriptions is found in large-deviations scaling [1, 2] of the prob- abilities for fluctuations of averaged quantities. The central function expressing large-deviations scaling is the entropy, which is the basis for both fluctuation theorems and for characterizing the thermodynamic interactions of systems. Freidlin-Wentzell theory [3] provides a quite general for- mulation of large-deviations scaling for non-equilibrium stochastic processes, through a remarkable representation in terms of a Hamiltonian dynamical system. A number of related methods now exist to construct the Freidlin-Wentzell Hamiltonian for many kinds of stochastic processes; one method due to Doi [4, 5] and Peliti [6, 7], appropriate to integer counting statistics, is widely used in reaction-diffusion theory. Using these tools together with a path-entropy method due to Jaynes [8], this review shows how to construct entropy functions that both express large-deviations scaling of fluctuations, and describe system-environment interactions, for discrete stochastic processes either at or away from equilibrium. A collection of variational methods familiar within quantum field theory, but less commonly applied to the Doi-Peliti construction, is used to define a “stochastic effective action”, which is the large-deviations rate function for arbitrary non-equilibrium paths. We show how common principles of entropy maximization, applied to different ensembles of states or of histories, lead to different entropy functions and different sets of thermodynamic state variables.
  • Momentum-Dependent Mass and AC Hall Conductivity of Quantum Anomalous Hall Insulators and Their Relation to the Parity Anomaly

    Momentum-Dependent Mass and AC Hall Conductivity of Quantum Anomalous Hall Insulators and Their Relation to the Parity Anomaly

    PHYSICAL REVIEW RESEARCH 2, 033193 (2020) Momentum-dependent mass and AC Hall conductivity of quantum anomalous Hall insulators and their relation to the parity anomaly Christian Tutschku, Jan Böttcher , René Meyer, and E. M. Hankiewicz* Institute for Theoretical Physics and Würzburg-Dresden Cluster of Excellence ct.qmat, Julius-Maximilians-Universität Würzburg, 97074 Würzburg, Germany (Received 9 March 2020; accepted 15 July 2020; published 4 August 2020) The Dirac mass of a two-dimensional QAH insulator is directly related to the parity anomaly of planar quantum electrodynamics, as shown initially by Niemi and Semenoff [Phys. Rev. Lett. 51, 2077 (1983)]. In this work, we connect the additional momentum-dependent Newtonian mass term of a QAH insulator to the parity anomaly. We reveal that in the calculation of the effective action, before renormalization, the Newtonian mass acts similar to a parity-breaking element of a high-energy regularization scheme. This calculation allows us to derive the finite frequency correction to the DC Hall conductivity of a QAH insulator. We predict that the leading order AC correction contains a term proportional to the Chern number. This term originates from the Newtonian mass and can be measured via electrical or magneto-optical experiments. Moreover, we prove that the Newtonian mass significantly changes the resonance structure of the AC Hall conductivity in comparison to pure Dirac systems like graphene. DOI: 10.1103/PhysRevResearch.2.033193 I. INTRODUCTION While the contribution of the Dirac mass to the effective ac- tion of a QED + systems was initially shown in Refs. [22,23], The discovery of Dirac materials, such as topological 2 1 the effective action of Chern insulator in the presence of a insulators [1–10] and Weyl or Dirac [11–16] semimetals Newtonian mass has not yet been analyzed.
  • 25 Years of Quantum Hall Effect

    25 Years of Quantum Hall Effect

    S´eminaire Poincar´e2 (2004) 1 – 16 S´eminaire Poincar´e 25 Years of Quantum Hall Effect (QHE) A Personal View on the Discovery, Physics and Applications of this Quantum Effect Klaus von Klitzing Max-Planck-Institut f¨ur Festk¨orperforschung Heisenbergstr. 1 D-70569 Stuttgart Germany 1 Historical Aspects The birthday of the quantum Hall effect (QHE) can be fixed very accurately. It was the night of the 4th to the 5th of February 1980 at around 2 a.m. during an experiment at the High Magnetic Field Laboratory in Grenoble. The research topic included the characterization of the electronic transport of silicon field effect transistors. How can one improve the mobility of these devices? Which scattering processes (surface roughness, interface charges, impurities etc.) dominate the motion of the electrons in the very thin layer of only a few nanometers at the interface between silicon and silicon dioxide? For this research, Dr. Dorda (Siemens AG) and Dr. Pepper (Plessey Company) provided specially designed devices (Hall devices) as shown in Fig.1, which allow direct measurements of the resistivity tensor. Figure 1: Typical silicon MOSFET device used for measurements of the xx- and xy-components of the resistivity tensor. For a fixed source-drain current between the contacts S and D, the potential drops between the probes P − P and H − H are directly proportional to the resistivities ρxx and ρxy. A positive gate voltage increases the carrier density below the gate. For the experiments, low temperatures (typically 4.2 K) were used in order to suppress dis- turbing scattering processes originating from electron-phonon interactions.
  • Arxiv:2002.11085V1 [Hep-Th]

    Arxiv:2002.11085V1 [Hep-Th]

    On-Shell Electric-Magnetic Duality and the Dual Graviton 1,2 2 Nathan Moynihan and Jeff Murugan ∗ 1High Energy Physics, Cosmology & Astrophysics Theory group, 2The Laboratory for Quantum Gravity & Strings Department of Mathematics and Applied Mathematics, University of Cape Town, Rondebosch, Cape Town 7700, South Africa Using on-shell amplitude methods, we explore 4-dimensional Electric-Magnetic duality and its double copy. We show explicitly that the on-shell scattering amplitudes know about ‘dual’ photons (and dual gravitons), that the off-shell photon propagator double copies to the graviton propagator and that the magnetic part of the propagator is essential for the double copy to hold. We also show that there is an equivalent gravito-magnetic part of the graviton propagator which is essential in giving rise to solutions with either angular momentum or NUT charge. Furthermore, we comment on the so-called Weinberg paradox, which states that scattering amplitudes involving the mixing of electric and magnetic monopoles cannot be Lorentz invariant, and would seem to preclude the existence of the ’t Hooft-Polyakov (topological) monopole. We trace this paradox to the magnetic part of the propagator, showing that it can be eliminated if one restricts to proper orthochronous Lorentz transformations. Finally, we compute the fully relativistic cross-section for arbitrary spin dyons using the recently formulated on-shell duality transformation and show that this is always fully Lorentz invariant. INTRODUCTION field theory without a Dirac string singularity necessitates the introduction of a second four-vector potential: the The boostrap program of the 1960’s received considerable dual photon [3–6].
  • SKYRMIONS and the V = 1 QUANTUM HALL FERROMAGNET

    SKYRMIONS and the V = 1 QUANTUM HALL FERROMAGNET

    o 9 (1997 ACA YSICA OŁOICA A Νo oceeigs o e I Ieaioa Scoo o Semicoucig Comous asowiec 1997 SKYMIOS A E v = 1 QUAUM A EOMAGE M MAA GOEG e o ysics oso Uiesiy oso MA 15 USA EIE A K WES e aoaoies uce ecoogies Muay i 797 USA ece eeimea a eoeica iesigaios ae esue i a si i ou uesaig o e v = 1 quaum a sae ee ow eiss a wea o eiece a e eciaio ga a e esuig quasiaice secum a v = 1 ae ue prdntl o e eomageic may-oy ecage ieacio A gea aiey o eeimeay osee coeaios a v = 1 nnt e icooae io a euaie easio aou e sige-aice moe a sceme og oug o escie e iega qua- um a eec a iig aco 1 eoiss ow ee o e v = 1 sae as e quaum a eomage I is ae we eiew ece eoei- ca a eeimea ogess a eai ou ow oica iesigaios o e v = 1 quaum a egime e ecique o mageo-asoio sec- oscoy as oe o e oweu a oe o e occuacy o e owes aau ee i e egime o 7 v 13 aou e si ga Aiioay we ae eome simuaeous measuemes o e asoio oou- miescece a ooumiescece eciaio seca o e v = 1 sae i oe o euciae e oe o ecioic a eaaio eecs i oica secoscoy i e quaum a egime ACS umes 73Ηm 7-w 73Mí 717Gm 73 1 Ioucio Some iee yeas ae e iscoey o e acioa quaum a e- ec [1] e suy o woimesioa eeco sysems (ES coie o e owes aau ee ( coiues o e a eie aoaoy o e iesiga- io o may-oy ieacios e oieaio o eeimeay osee ac- ioa a saes [] e iscoey o comosie emios a a-iig [3] a e ossiiiy o eoic si-uoaie acioa gou saes [] ae u a ew eames o e aomaies a coiue o caege ou uesaig o sog eeco—eeco coeaios i e eeme mageic quaum imi ecey e v = 1 quaum a sae a egio oug o e we-uesoo wii e (1 622 M.J.
  • Large N Free Energy of 3D N=4 Scfts and Ads/CFT Benjamin Assel, John Estes, Masahito Yamazaki

    Large N Free Energy of 3D N=4 Scfts and Ads/CFT Benjamin Assel, John Estes, Masahito Yamazaki

    Large N Free Energy of 3d N=4 SCFTs and AdS/CFT Benjamin Assel, John Estes, Masahito Yamazaki To cite this version: Benjamin Assel, John Estes, Masahito Yamazaki. Large N Free Energy of 3d N=4 SCFTs and AdS/CFT. Journal of High Energy Physics, Springer, 2012, pp.074. 10.1007/JHEP09(2012)074. hal-00718824 HAL Id: hal-00718824 https://hal.archives-ouvertes.fr/hal-00718824 Submitted on 18 Jul 2012 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. LPTENS-12/23 PUPT-2418 Large N Free Energy of 3d = 4 SCFTs and N AdS4/CFT3 Benjamin Assel\, John Estes[ and Masahito Yamazaki] \ Laboratoire de Physique Th´eoriquede l'Ecole´ Normale Sup´erieure, 24 rue Lhomond, 75231 Paris cedex, France [ Instituut voor Theoretische Fysica, KU Leuven, Celestijnenlaan 200D B-3001 Leuven, Belgium ] Princeton Center for Theoretical Science, Princeton University, Princeton, NJ 08544, USA Abstract We provide a non-trivial check of the AdS4/CFT3 correspondence recently proposed in [1] by verifying the GKPW relation in the large N limit. The CFT free energy is obtained from the previous works [2, 3] on the S3 partition function for 3-dimensional = 4 SCFT T [SU(N)].
  • Ads₄/CFT₃ and Quantum Gravity

    Ads₄/CFT₃ and Quantum Gravity

    AdS/CFT and quantum gravity Ioannis Lavdas To cite this version: Ioannis Lavdas. AdS/CFT and quantum gravity. Mathematical Physics [math-ph]. Université Paris sciences et lettres, 2019. English. NNT : 2019PSLEE041. tel-02966558 HAL Id: tel-02966558 https://tel.archives-ouvertes.fr/tel-02966558 Submitted on 14 Oct 2020 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Prepar´ ee´ a` l’Ecole´ Normale Superieure´ AdS4/CF T3 and Quantum Gravity Soutenue par Composition du jury : Ioannis Lavdas Costas BACHAS Le 03 octobre 2019 Ecole´ Normale Superieure Directeur de These Guillaume BOSSARD Ecole´ Polytechnique Membre du Jury o Ecole´ doctorale n 564 Elias KIRITSIS Universite´ Paris-Diderot et Universite´ de Rapporteur Physique en ˆIle-de-France Crete´ Michela PETRINI Sorbonne Universite´ President´ du Jury Nicholas WARNER University of Southern California Membre du Jury Specialit´ e´ Alberto ZAFFARONI Physique Theorique´ Universita´ Milano-Bicocca Rapporteur Contents Introduction 1 I 3d N = 4 Superconformal Theories and type IIB Supergravity Duals6 1 3d N = 4 Superconformal Theories7 1.1 N = 4 supersymmetric gauge theories in three dimensions..............7 1.2 Linear quivers and their Brane Realizations...................... 10 1.3 Moduli Space and Symmetries............................
  • Comparison Between Quantum and Classical Dynamics in the Effective Action Formalism

    Comparison Between Quantum and Classical Dynamics in the Effective Action Formalism

    Proceedings of the International School of Physics "Enrico Fermi" Course CXLIII. G. Casati, I. Guarneri and U. Smilansky (Eds.) lOS Press, Amsterdam 2000 Comparison between quantum and classical dynamics in the effective action formalism F. CAMETTI Dipartimento di Fisica, Universitd della Calabria - 87036 Arcavacata di Rende, Italy and Istituto Nazionale per la Fisica della Materia, unita di Cosenza - Italy G. JONA-LASINIO Dipartimento di Fisica, Universita di Roma "La Sapienza"- P.le A. Moro 2, 00185 Roma, Italy C. PRESILLA Dipartimento di Fisica, Universita di Roma "La Sapienza"- P.le A. Moro 2, 00185 Roma, Italy and Istituto Nazionale per la Fisica della Materia, unita di Roma "La Sapienza" - Italy F. TONINELLI Dipartimento di Fisica, Universita di Roma "La Sapienza"- P.le A. Moro 2, 00185 Roma, Italy A major difficulty in comparing quantum and classical behavior resides in the struc­ tural differences between the corresponding mathematical languages. The Heisenberg equations of motion are operator equations only formally identical to the classical equa­ tions of motion. By taking the expectation of these equations, the well-known Ehrenfest theorem provides identities which, however, are not a closed system of equations which allows to evaluate the time evolution of the system. The formalism of the effective action seems to offer a possibility of comparing quantum and classical evolutions in a system­ atic and logically consistent way by naturally providing approximation schemes for the expectations of the coordinates which at the zeroth order coincide with the classical evolution [1]. The effective action formalism leads to equations of motion which differ from the classical equations by the addition of terms nonlocal in the time variable.
  • Recent Experimental Progress of Fractional Quantum Hall Effect: 5/2 Filling State and Graphene

    Recent Experimental Progress of Fractional Quantum Hall Effect: 5/2 Filling State and Graphene

    Recent Experimental Progress of Fractional Quantum Hall Effect: 5/2 Filling State and Graphene X. Lin, R. R. Du and X. C. Xie International Center for Quantum Materials, Peking University, Beijing, People’s Republic of China 100871 ABSTRACT The phenomenon of fractional quantum Hall effect (FQHE) was first experimentally observed 33 years ago. FQHE involves strong Coulomb interactions and correlations among the electrons, which leads to quasiparticles with fractional elementary charge. Three decades later, the field of FQHE is still active with new discoveries and new technical developments. A significant portion of attention in FQHE has been dedicated to filling factor 5/2 state, for its unusual even denominator and possible application in topological quantum computation. Traditionally FQHE has been observed in high mobility GaAs heterostructure, but new materials such as graphene also open up a new area for FQHE. This review focuses on recent progress of FQHE at 5/2 state and FQHE in graphene. Keywords: Fractional Quantum Hall Effect, Experimental Progress, 5/2 State, Graphene measured through the temperature dependence of I. INTRODUCTION longitudinal resistance. Due to the confinement potential of a realistic 2DEG sample, the gapped QHE A. Quantum Hall Effect (QHE) state has chiral edge current at boundaries. Hall effect was discovered in 1879, in which a Hall voltage perpendicular to the current is produced across a conductor under a magnetic field. Although Hall effect was discovered in a sheet of gold leaf by Edwin Hall, Hall effect does not require two-dimensional condition. In 1980, quantum Hall effect was observed in two-dimensional electron gas (2DEG) system [1,2].
  • Introduction to Effective Field Theory

    Introduction to Effective Field Theory

    Effective Field Theories 1 Introduction to Effective Field Theory C.P. Burgess Department of Physics & Astronomy, McMaster University, 1280 Main Street West, Hamilton, Ontario, Canada, L8S 4M1 and Perimeter Institute, 31 Caroline Street North, Waterloo, Ontario, Canada, N2L 2Y5 Key Words Effective field theory; Low-energy approximation; Power-counting Abstract This review summarizes Effective Field Theory techniques, which are the modern theoretical tools for exploiting the existence of hierarchies of scale in a physical problem. The general theoretical framework is described, and explicitly evaluated for a simple model. Power- counting results are illustrated for a few cases of practical interest, and several applications to arXiv:hep-th/0701053v2 18 Jan 2007 Quantum Electrodynamics are described. CONTENTS Introduction ....................................... 2 A toy model .......................................... 3 General formulation ................................... 6 The 1PI and 1LPI actions .................................. 6 The Wilson action ...................................... 20 Ann. Rev. Nucl. Part. Sci. 2000 1056-8700/97/0610-00 Power counting ...................................... 28 A class of effective interactions ............................... 28 Power-counting rules ..................................... 29 The Effective-Lagrangian Logic ............................... 32 Applications ....................................... 34 Quantum Electrodynamics .................................. 34 Power-counting examples: QCD and Gravity ....................... 46 1 Introduction It is a basic fact of life that Nature comes to us in many scales. Galaxies, planets, aardvarks, molecules, atoms and nuclei are very different sizes, and are held together with very different binding energies. Happily enough, it is another fact of life that we don’t need to understand what is going on at all scales at once in order to figure out how Nature works at a particular scale. Like good musicians, good physicists know which scales are relevant for which compositions.