Aspects of the Quantum Hall Effect
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The Second Quantized Approach to the Study of Model Hamiltonians in Quantum Hall Regime
Washington University in St. Louis Washington University Open Scholarship Arts & Sciences Electronic Theses and Dissertations Arts & Sciences Winter 12-15-2016 The Second Quantized Approach to the Study of Model Hamiltonians in Quantum Hall Regime Li Chen Washington University in St. Louis Follow this and additional works at: https://openscholarship.wustl.edu/art_sci_etds Recommended Citation Chen, Li, "The Second Quantized Approach to the Study of Model Hamiltonians in Quantum Hall Regime" (2016). Arts & Sciences Electronic Theses and Dissertations. 985. https://openscholarship.wustl.edu/art_sci_etds/985 This Dissertation is brought to you for free and open access by the Arts & Sciences at Washington University Open Scholarship. It has been accepted for inclusion in Arts & Sciences Electronic Theses and Dissertations by an authorized administrator of Washington University Open Scholarship. For more information, please contact [email protected]. WASHINGTON UNIVERSITY IN ST. LOUIS Department of Physics Dissertation Examination Committee: Alexander Seidel, Chair Zohar Nussinov Michael Ogilvie Xiang Tang Li Yang The Second Quantized Approach to the Study of Model Hamiltonians in Quantum Hall Regime by Li Chen A dissertation presented to The Graduate School of Washington University in partial fulfillment of the requirements for the degree of Doctor of Philosophy December 2016 Saint Louis, Missouri c 2016, Li Chen ii Contents List of Figures v Acknowledgements vii Abstract viii 1 Introduction to Quantum Hall Effect 1 1.1 The classical Hall effect . .1 1.2 The integer quantum Hall effect . .3 1.3 The fractional quantum Hall effect . .7 1.4 Theoretical explanations of the fractional quantum Hall effect . .9 1.5 Laughlin's gedanken experiment . -
Magnetism, Free Electrons and Interactions
Magnetism Magnets Zero external field Finite external field • Types of magnetic systems • Pauli paramagnetism in metals Paramagnets • Landau diamagnetism in metals • Larmor diamagnetism in insulators Diamagnets • Ferromagnetism of electron gas • Spin Hamiltonian Ferromagnets • Mean field approach • Curie transition Antiferromagnets Ferrimagnets …… … Pauli paramagnetism Pauli paramagnetism Let us first look at magnetic properties of a free electron gas. ε =−p2 /2meBmc= /2 ε =+p2 /2meBmc= /2 ↑ ↓G Electron are spin-1/2 particles #of majority spins: dp3 NV= f()ε In external magnetic field B – Zeeman splitting #of minority spins: ↑,,↓ ∫ (2π= )3 ↑ ↓ 2 = 2 = ε↑ =−p /2meBmc /2 ε↓ =+p /2meBmc /2 Magnetization (magnetic moment per unit volume): - minority spins e= M =−()NN : aligned along the field and proportional Fermi level ↑ ↓ 2Vmc to B in low fields χ - magnetic succeptibility - majority spins M = χB χ > 0 - paramagnetism Pauli succeptibility Landau quantization G 2 2 A free electron in magnetic field: B & zˆ ε↑ =−p /2meBmc= /2 ε↓ =+p /2meBmc= /2 G 2 µ+eB= /2 mc 2 G VgeB= Schrödinger equation: = ⎛⎞ieA ABxAA===;0 NN−= gd()εε ≈ V −∇+=⎜⎟ψ εψ yxz ↑↓ ∫ 2mc= 22µ−eB= /2 mc mc ⎝⎠ B=1T corresponds toeBmc= /1=× K k provided m is free electrons’s mass Solutions: labeled by two indices nk, B G z For any fields, eBmc=/ µ ψ nk(r )= exp( ik y y+− ik z z )ϕ n ( x= ck y / eB ) Magnetic succeptibility: ϕn - wave functions of a harmonic oscillator 22 2 Energies: ε =+==kmeBmcn/2 ( / )( + 1/2) - strongly degenerate!! ⎛⎞e= nk z χP = ⎜⎟g ⎝⎠2mc We “quantized” momenta transverse to the field (Landau levels) 1 Landau diamagnetism Electrons in metals A free electron in magnetic field: moves along spiral trajectories We know that there are diamagnetic metals. -
Canonical Quantization of Two-Dimensional Gravity S
Journal of Experimental and Theoretical Physics, Vol. 90, No. 1, 2000, pp. 1–16. Translated from Zhurnal Éksperimental’noœ i Teoreticheskoœ Fiziki, Vol. 117, No. 1, 2000, pp. 5–21. Original Russian Text Copyright © 2000 by Vergeles. GRAVITATION, ASTROPHYSICS Canonical Quantization of Two-Dimensional Gravity S. N. Vergeles Landau Institute of Theoretical Physics, Russian Academy of Sciences, Chernogolovka, Moscow oblast, 142432 Russia; e-mail: [email protected] Received March 23, 1999 Abstract—A canonical quantization of two-dimensional gravity minimally coupled to real scalar and spinor Majorana fields is presented. The physical state space of the theory is completely described and calculations are also made of the average values of the metric tensor relative to states close to the ground state. © 2000 MAIK “Nauka/Interperiodica”. 1. INTRODUCTION quantization) for observables in the highest orders, will The quantum theory of gravity in four-dimensional serve as a criterion for the correctness of the selected space–time encounters fundamental difficulties which quantization route. have not yet been surmounted. These difficulties can be The progress achieved in the construction of a two- arbitrarily divided into conceptual and computational. dimensional quantum theory of gravity is associated The main conceptual problem is that the Hamiltonian is with two ideas. These ideas will be formulated below a linear combination of first-class constraints. This fact after the necessary notation has been introduced. makes the role of time in gravity unclear. The main computational problem is the nonrenormalizability of We shall postulate that space–time is topologically gravity theory. These difficulties are closely inter- equivalent to a two-dimensional cylinder. -
Introduction to Quantum Field Theory
Utrecht Lecture Notes Masters Program Theoretical Physics September 2006 INTRODUCTION TO QUANTUM FIELD THEORY by B. de Wit Institute for Theoretical Physics Utrecht University Contents 1 Introduction 4 2 Path integrals and quantum mechanics 5 3 The classical limit 10 4 Continuous systems 18 5 Field theory 23 6 Correlation functions 36 6.1 Harmonic oscillator correlation functions; operators . 37 6.2 Harmonic oscillator correlation functions; path integrals . 39 6.2.1 Evaluating G0 ............................... 41 6.2.2 The integral over qn ............................ 42 6.2.3 The integrals over q1 and q2 ....................... 43 6.3 Conclusion . 44 7 Euclidean Theory 46 8 Tunneling and instantons 55 8.1 The double-well potential . 57 8.2 The periodic potential . 66 9 Perturbation theory 71 10 More on Feynman diagrams 80 11 Fermionic harmonic oscillator states 87 12 Anticommuting c-numbers 90 13 Phase space with commuting and anticommuting coordinates and quanti- zation 96 14 Path integrals for fermions 108 15 Feynman diagrams for fermions 114 2 16 Regularization and renormalization 117 17 Further reading 127 3 1 Introduction Physical systems that involve an infinite number of degrees of freedom are usually described by some sort of field theory. Almost all systems in nature involve an extremely large number of degrees of freedom. A droplet of water contains of the order of 1026 molecules and while each water molecule can in many applications be described as a point particle, each molecule has itself a complicated structure which reveals itself at molecular length scales. To deal with this large number of degrees of freedom, which for all practical purposes is infinite, we often regard a system as continuous, in spite of the fact that, at certain distance scales, it is discrete. -
Rich Magnetic Quantization Phenomena in AA Bilayer Silicene
www.nature.com/scientificreports OPEN Rich Magnetic Quantization Phenomena in AA Bilayer Silicene Po-Hsin Shih1, Thi-Nga Do2,3, Godfrey Gumbs4,5, Danhong Huang6, Hai Duong Pham1 & Ming-Fa Lin1,7,8 Received: 13 June 2019 The rich magneto-electronic properties of AA-bottom-top (bt) bilayer silicene are investigated using Accepted: 27 August 2019 a generalized tight-binding model. The electronic structure exhibits two pairs of oscillatory energy Published: xx xx xxxx bands for which the lowest conduction and highest valence states of the low-lying pair are shifted away from the K point. The quantized Landau levels (LLs) are classifed into various separated groups by the localization behaviors of their spatial distributions. The LLs in the vicinity of the Fermi energy do not present simple wave function modes. This behavior is quite diferent from other two-dimensional systems. The geometry symmetry, intralayer and interlayer atomic interactions, and the efect of a perpendicular magnetic feld are responsible for the peculiar LL energy spectra in AA-bt bilayer silicene. This work provides a better understanding of the diverse magnetic quantization phenomena in 2D condensed-matter materials. Silicene, an isostructure to graphene, is purely made of silicon atoms through both the sp2 and sp3 bondings. So far, silicene systems have been successfully synthesized by the epitaxial growth on various substrate surfaces. Monolayer silicene with diferent sizes of unit cells has been produced on several substrates, such as Si(111) 1,2 3,4 5 6 33× -Ag template , Ag(111) (4 × 4) , Ir(111) ( 33× ) and ZrB2(0001) (2 × 2) . -
Topological Classification of Correlations in 2D Electron
materials Article Topological Classification of Correlations in 2D Electron Systems in Magnetic or Berry Fields Janusz E. Jacak Department of Quantum Technologies, Wrocław University of Science and Technology, Wyb. Wyspia´nskiego27, 50-370 Wrocław, Poland; [email protected] Abstract: Recent topology classification of 2D electron states induced by different homotopy classes of mappings of the planar Brillouin zone into Bloch space can be supplemented by a homotopy classification of various phases of multi-electron homotopy patterns induced by Coulomb interaction between electrons. The general classification of such type is presented. It explains the topologically protected correlations responsible for integer and fractional Hall effects in 2D multi-electron systems in the presence of perpendicular quantizing magnetic field or Berry field, the latter in topological Chern insulators. The long-range quantum entanglement is essential for homotopy correlated phases in contrast to local binary entanglement for conventional phases with local order parameters. The classification of homotopy long-range correlated phases induced by the Coulomb interaction of electrons has been derived in terms of homotopy invariants and illustrated by experimental observations in GaAs 2DES, graphene monolayer, and bilayer and in Chern topological insulators. The homotopy phases are demonstrated to be topologically protected and immune to the local crystal field, local disorder, and variation of the electron interaction strength. The nonzero interaction between electrons is shown, however, to be essential for the definition of the homotopy invariants, which disappear in gaseous systems. Citation: Jacak, J.E. Topological Keywords: homotopy phases; long-range quantum entanglement; FQHE; Hall systems; Chern Classification of Correlations in 2D topological insulators Electron Systems in Magnetic or Berry Field. -
Composite Fermion Theory of Excitations in the Fractional Quantum Hall Effect
Solid State Communications 135 (2005) 602–609 www.elsevier.com/locate/ssc Composite fermion theory of excitations in the fractional quantum Hall effect J.K. Jaina,*, K. Parkb, M.R. Petersona, V.W. Scarolab a104 Davey Laboratory, Physics Department, The Pennsylvania State University, University Park, PA 16802, USA bPhysics Department, Condensed Matter Theory Center, University of Maryland, College Park, MD 20742, USA Received 28 February 2005; accepted 1 May 2005 by the guest editors Available online 13 May 2005 Abstract Transport experiments are sensitive to charged ‘quasiparticle’ excitations of the fractional quantum Hall effect. Inelastic Raman scattering experiments have probed an amazing variety of other excitations: excitons, rotons, bi-rotons, trions, flavor altering excitons, spin waves, spin-flip excitons, and spin-flip rotons. This paper reviews the status of our theoretical understanding of these excitations. q 2005 Elsevier Ltd. All rights reserved. PACS: 71.10.Pm; 73.43.Kf Keywords: A. Composite fermion; D. Fractional quantum Hall effects 1. Introduction have revealed a strikingly rich structure with many kinds of excitations. Transport experiments told us quite early on that there is There has also been substantial theoretical progress on a gap to charged excitations in the fractional quantum Hall this issue, and an excellent description of the neutral effect (FQHE) [1]. The longitudinal resistance was seen to excitations has been achieved in terms of a particle hole pair behave like rxxZexp(KD/2kBT) in a range of temperature, of composite fermions (CFs), which dovetails nicely with and the quantity D is interpreted as the minimum energy our understanding of the incompressible ground states as an required to create a charged excitation. -
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. -
Path-Memory Induced Quantization of Classical Orbits SEE COMMENTARY
Path-memory induced quantization of classical orbits SEE COMMENTARY Emmanuel Forta,1, Antonin Eddib, Arezki Boudaoudc, Julien Moukhtarb, and Yves Couderb aInstitut Langevin, Ecole Supérieure de Physique et de Chimie Industrielles ParisTech and Université Paris Diderot, Centre National de la Recherche Scientifique Unité Mixte de Recherche 7587, 10 Rue Vauquelin, 75 231 Paris Cedex 05, France; bMatières et Systèmes Complexes, Université Paris Diderot, Centre National de la Recherche Scientifique Unité Mixte de Recherche 7057, Bâtiment Condorcet, 10 Rue Alice Domon et Léonie Duquet, 75013 Paris, France; and cLaboratoire de Physique Statistique, Ecole Normale Supérieure, 24 Rue Lhomond, 75231 Paris Cedex 05, France Edited* by Pierre C. Hohenberg, New York University, New York, NY, and approved August 4, 2010 (received for review May 26, 2010) A droplet bouncing on a liquid bath can self-propel due to its inter- a magnetic field. However, for technical reasons we chose a action with the waves it generates. The resulting “walker” is a variant that relies on the analogy first introduced by Berry et al. dynamical association where, at a macroscopic scale, a particle (5) between electromagnetic fields and surface waves. ~ ~ ~ (the droplet) is driven by a pilot-wave field. A specificity of this Its starting point is the similarity of relation B ¼ ∇ × A in elec- ~ system is that the wave field itself results from the superposition tromagnetism with 2Ω~¼ ∇~× U in fluid mechanics. In these re- ~ of the waves generated at the points of space recently visited by lations, the vorticity 2Ω~is the equivalent of the magnetic field B ~ ~ the particle. It thus contains a memory of the past trajectory of the and the velocity U that of the vector potential A. -
Quantum Droplets in Dipolar Ring-Shaped Bose-Einstein Condensates
UNIVERSITY OF BELGRADE FACULTY OF PHYSICS MASTER THESIS Quantum Droplets in Dipolar Ring-shaped Bose-Einstein Condensates Author: Supervisor: Marija Šindik Dr. Antun Balaž September, 2020 UNIVERZITET U BEOGRADU FIZICKIˇ FAKULTET MASTER TEZA Kvantne kapljice u dipolnim prstenastim Boze-Ajnštajn kondenzatima Autor: Mentor: Marija Šindik Dr Antun Balaž Septembar, 2020. Acknowledgments I would like to express my sincerest gratitude to my advisor Dr. Antun Balaž, for his guidance, patience, useful suggestions and for the time and effort he invested in the making of this thesis. I would also like to thank my family and friends for the emotional support and encouragement. This thesis is written in the Scientific Computing Laboratory, Center for the Study of Complex Systems of the Institute of Physics Belgrade. Numerical simulations were run on the PARADOX supercomputing facility at the Scientific Computing Laboratory of the Institute of Physics Belgrade. iii Contents Chapter 1 – Introduction 1 Chapter 2 – Theoretical description of ultracold Bose gases 3 2.1 Contact interaction..............................4 2.2 The Gross-Pitaevskii equation........................5 2.3 Dipole-dipole interaction...........................8 2.4 Beyond-mean-field approximation......................9 2.5 Quantum droplets............................... 13 Chapter 3 – Numerical methods 15 3.1 Rescaling of the effective GP equation................... 15 3.2 Split-step Crank-Nicolson method...................... 17 3.3 Calculation of the ground state....................... 20 3.4 Calculation of relevant physical quantities................. 21 Chapter 4 – Results 24 4.1 System parameters.............................. 24 4.2 Ground state................................. 25 4.3 Droplet formation............................... 26 4.4 Critical strength of the contact interaction................. 28 4.5 The number of droplets........................... -
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. -
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].