Quantum Hall Effects

The following sections are included:

• PREFACE TO THE THIRD EDITION

• PREFACE TO THE SECOND EDITION

• PREFACE TO THE FIRST EDITION

• CONTENTS

The following sections are included:

• Hilbert Spaces

• Canonical Formalism

• Creation and Annihilation Operators

• Uncertainty Principle

• Coherent States and von Neumann Lattice

• Squeezed Coherent State

• Particle Number and Phase

• Macroscopic Coherence

The following sections are included:

• One-Body Hamiltonian

• Many-Body Hamiltonian

• Boson Field Operators

• Quantum Field Theory

• Fermion Field Operators

• Electrons Interacting with Electromagnetic Field

The following sections are included:

• Relativistic Particles and Waves

• Schrödinger Field

• Real Klein-Gordon Field

• Complex Klein-Gordon Field

• Nöther Currents

The following sections are included:

• Ferromagnets

• Real Klein-Gordon Field

• Complex Klein-Gordon Field

• Sigma Model

• Spin Field

• Schrödinger Field

• Superfluidity

• Goldstone Theorem

The following sections are included:

• Maxwell Equations

• Canonical Quantization

• Interaction with Matter Field

• Massive Vector Field

• Superconductivity

• Aharonov-Bohm Effect

The following sections are included:

• Dirac Equation

• Plane Wave Solutions

• Canonical Quantization

• Interaction with Electromagnetic Field

• Weyl Field (Massless Dirac Field)

• Massless Dirac Electrons in Magnetic Field

The following sections are included:

• Topological Sectors

• Classical Fields

• Solitary Waves, Kinks and Solitons

• Sine-Gordon Solitons

• Vortex Solitons

• Homotopy Classes

• O(3) Skyrmions

• SU(N) Skyrmions or CP^N-1 Skyrmions

• Skyrmions in Ferromagnets

The following sections are included:

• Spin and Statistics

• Fractional Statistics

• Quantum Mechanics

• Chern-Simons Gauge Theory

• Anyon Field Operators

The quantum Hall (QH) effect is one of the most remarkable phenomena discovered in the last century. The integer QH effect was discovered in 1980 by Klaus von Klitzing, while the fractional QH effect was discovered in 1982 by Daniel Tsui, Horst Strömer and Arthur Gossard. It was predicted by Robert Laughlin that a quasiparticle is an anyon carrying electric charge e/m at the filling factor ν =1/m. The QH state forms an incompressible liquid, which is best understood in terms of composite particles. The ground state is a QH ferromagnet at ν =1, where charged excitations are topological solitons identified with skyrmions. On the other hand, the edge forms a chiral Tomonaga-Luttinger liquid. A recent topic is the discovery of an unconventional QH effect in graphene, where electrons are described by the Dirac equation rather than the Schrö dinger equation. Another topic is the anomolous QH effect, which is the QH effect without Landau levels. It blazed a way to the paradigm of topological insulators. In this section we overview various aspects of the QH effect.

The following sections are included:

• Planar Electrons

• Cyclotron Motion

• Symmetric Gauge

• Landau Gauge

• von Neumann Lattice

• Electrons in Nth Landau Level

• Hall Current

The following sections are included:

• Incompressibility

• Integer Quantum Hall Effects

• Fractional Quantum Hall Effects

• Quasiparticles

• Hall Plateaux

The following sections are included:

• Impurity Potentials

• Gap Energies

• Dispersion Relation

• Thermal Activation

The following sections are included:

• Composite Particles

• Statistical Transmutation

• Effective Magnetic Field

• Dressed Composite Particles

• Composite Particles in Lowest Landau Level

• Composite Fermions in Lowest Landau Level

The following sections are included:

• Ground State and Laughlin Wave Function

• Perturbative Excitations

• Vortex Excitations

• Field Theory of Vortex Solitons

• Haldane-Halperin Hierarchy

The following sections are included:

• Spin Coherence

• Spin Degree of Freedom

• Composite Bosons and Spin-Charge Separation

• Spin Field, Sigma Field and CP¹ Field

• Effective Hamiltonian

The following sections are included:

• Spin Excitations

• Factorizable Skyrmions

• Skyrmion Excitation Energy

• Experimental Evidence

The following sections are included:

• Jain Hierarchy

• Landau Levels of Composite Fermions

• Beyond Principal Sequences

• Spin Polarization

• Gap Energies

The following sections are included:

• Edge and Bulk

• Shot Noises of Fractional Charges

• Chiral Edge Excitations

• Chiral Tomonaga-Luttinger Liquid

• Electrodynamics on Edge

• Edge Tunneling and Sine-Gordon Solitons

Electrons occupy only the lowest Landau level in the limit of high magnetic field. With decreasing magnetic field, the number of degeneracy in each Landau level decreases and part of electrons occupy higher Landau levels. Since the Coulomb potential is not a monotonically decreasing function in higher Landau levels, we may expect charge density waves (CDWs) such as stripes and bubbles to emerge. In these CDW states electrons form clusters with other electrons, which contrasts to the Laughlin state realized in the lowest Landau level. The stripe phase has been observed experimentally by way of a strong anisotropy and nonlinearity in the diagonal resistivity R_xx.

An unconventional QH effect has been discovered in graphene, where electrons are described by the massless Dirac equation. The filling factors form a series ν = ±2, ±6, ±10, ⋯, implying the emergence of the zero-energy state and the 4-fold degeneracy of each energy level. The key observation is that the intrinsic Zeeman effect causes a Landau-level mixing to create the SUSY energy spectrum together with the zero-energy state inspite of the cyclotron motion of electrons. The Coulomb interaction becomes important when strong magnetic field is applied, and the graphene system may be regarded as an SU(4) QH ferromagnet. The series ν = 0, ±1, ±2, ±3, ⋯ of the full integers is expected to appear as a result of spontaneous symmetry breakdown.

Silicene is a monolayer of silicon atoms forming a two-dimensional honeycomb lattice, where electrons are Dirac fermions whose mass is experimentally controllable under electric field. We investigate quantum Hall effects in silicene by applying electric field E_z parallel to magnetic field. The 4-fold degenerate zero-energy states are completely resolved even without considering Coulomb interactions. The remarkable feature is that, by applying the electric field, we can control the valley degeneracy. As a function of E_z, Hall plateaux appear at the filling factors v = 0, ±1, ±2, ±3, ⋯ except for the points where level crossings occur.

The QH effect is associated with the presence of a uniform external magnetic field, which splits the spectrum of electron energy levels into Landau levels. In spite of this common sense it is possible to realize the QH effect without Landau levels on graphene or silicene. This is made possible by breaking the time-reversal symmetry with the use of off-resonance coherent laser beam or with the introduction of exchange effect. The QH effect without Landau levels is referred to as the quantum anomalous Hall effect. There occurs an analogous phenomenon, the quantum spin Hall effect, where a spin current flows instead of a charge current. They are typical phenomena on the topological insulator. We review various topological insulators materialized in the silicene system.

Bilayer systems exhibit a new variety of QH states. Various QH states are realized by controlling system parameters such as the strengths of the interlayer and intralayer Coulomb interactions, the tunneling interaction and the Zeeman effect. Here we overview diverse aspects of bilayer QH systems. A prominent feature is a spontaneous development of interlayer phase coherence driven by the interlayer exchange interaction. We explore a new type of the Josephson tunneling of the fundamental charge (-e) at ν = 1. We also investigate charge/spin Josephson supercurrents in the canted antiferromagnetic (CAF) phase at ν = 2. Quasiparticles are skyrmions, which may be viewed as a pair of merons in bilayer systems.

The pseudospin SU(2) structure is introduced into the bilayer system by assigning "up" and "down" pseudospins to the electrons on the front and back layers, respectively. The electron configuration is locally described by the total number density ρ(x) and the pseudospin density P(x). The bilayer QH state belongs to a pseudospin multiplet, where the pseudospin direction is controlled by applying bias voltages. The state belongs to a discrete spectrum or a continuous spectrum.

The following sections are included:

• Composite-Boson Field

• Wave Functions

• Ground State

• Vortex Excitations

The interlayer phase coherence is studied in the spin-frozen bilayer QH system. Each Landau level contains two energy levels with up and down pseudospins. These two levels are degenerate in the vanishing limit of the tunneling interaction, where pseudospins are spontaneously polarized to minimize the interlayer exchange energy. The pseudospin wave is a Goldstone mode having a linear dispersion relation in the vanishing limit of the tunneling gap △_SAS. It drives the Josephson current, which is dissipationless and leads to the Josephson-like effects between the two layers. They are manifested in the counterflow experiment, the drag experiment and other tunneling experiments.

The following sections are included:

• Bilayer QH Junction

• Parallel Magnetic Field

• Effective Hamiltonian

• Commensurate-Incommensurate Phase Transition

• Soliton Lattice

• Anomalous Diagonal Resistivity

• Josephson-like Phenomena in Parallel Field

The bilayerQHsystem has four energy levels in the lowest Landau level, corresponding to the layer and spin degrees of freedom. We investigate the system in the regime where all four levels are nearly degenerate and equally active, where the underlying group structure is SU(4). At ν = 1 the QH state is a charge-transferable state, and the SU(4) isospin coherence develops spontaneously. Quasiparticles are SU(4) skyrmions. One quasiparticle consists of a pair of charged excitations in the front and back layers. The SU(4) skyrmion evolves continuously from the ppin-skyrmion limit into the spin-skyrmion limit as the system is transformed from the balanced point to the monolayer point by controlling the bias voltage.

The following sections are included:

• Spin Phase, Ppin Phase and CAF Phase

• Ground-State Energy

• Ground-State Structure

• Phase Diagrams

• Experimental Data

• SU(4) Breaking and Goldstone Modes

• Grassmannian Fields and Goldstone modes

• Grassmannian G_4,2 Solitons

• Genuine Bilayer versus Two-Monolayer Systems

• Experimental Indication of Biskyrmions

Excitations across Landau levels are suppressed when the cyclotron energy is large enough. A self-consistent theory without these excitations is constructed by making the Landau-level projection. An electron confined to a single Landau level is described by the guiding center (X, Y) subject to noncommutativity, . Thus the QH system provides us with a realistic world of noncommutative geometry. It presents not only a good approximation but also an essential way to reveal a new physics inherent to the QH system. In formulating a microscopic theory of the integer QH system, we derive some key formulas already used in the semiclassical analysis. In this section we overview a microscopic theory of the QH effects based on noncommutative geometry. The complex number is defined by z =(x + iy) /ℓB throughout this Part.

The following sections are included:

• Noncommutative Coordinate

• Weyl Operator and Symbol

• Magnetic Translation

• Density Operators

• SU(N)-Extended W_∞ Algebra

• Classical Fields

• Topological Charge Density

• Kac-Moody Algebra on Edges

The following sections are included:

• Projected Coulomb Interactions

• Monolayer QH System

• Electron-Hole Pair Excitations

• Electron Excitation and Hole Excitation

• Bilayer System without Spin (ν = 1)

• Bilayer System with Spin (ν = 1)

• Bilayer System with Spins (ν = 2)

The following sections are included:

• Topological Charge and Electric Charge

• Microscopic Skyrmion States

• Noncommutative CP¹ Skyrmion

• Hole and Skyrmion

• Skyrmion Wave Functions

• Hardcore Interaction

• Coulomb Interaction

• Bilayer System in Parallel Magnetic Field

The following sections are included:

• Exchange Hamiltonian

• Decomposition Formula

• Spontaneous Symmetry Breaking

• Classical Equations of Motion

• Four-Layer Condenser Model

• Derivative Expansion and Effective Theory

• Noncommutative CP^N-1 Model

• Equations of Motion and Hall Currents

• Electric Currents in Pseudospin QH Ferromagnet

The experimental study of integer and fractional QH effects has a long history since their discoveries by Klaus von Klitzing in 1980 and Daniel C. Tsui, Horst Störmer and Arthur Gossard in 1982. This Part gives a glimpse into the broad and still flourishing field of QH research. The following chapters are devoted to review articles on some selected topics, provided by the experimentalists themselves. They will report on (i) the local-density-of-state patterns of QH states at nanometer scale, (ii) the excitation dispersions of magnetoplasmons, fractional-gap excitations, composite fermions, and magnetorotons, (iii) the exploitation of the hyperfine interaction between electron spin and nuclear spin for nanostructure devices, (iv) microwaveinduced zero resistance states, (v) a correlated state in bilayer QH systems with a Josephson-like tunneling effect, and (vi) QH effects in novel oxide materials.

Under a perpendicular magnetic field in two-dimensional electron systems, self-interference of the circular electronic orbits, i.e., the cyclotron orbits, leads to a standing wave pattern of the electronic probability density. This quantum pattern meanders along equipotential lines of the electrostatic potential disorder, which are in principal responsible for the integer quantum Hall effect. This microscopic picture motivates direct observation of the quantum Hall states by high resolution scanning tunneling spectroscopy.

This review is devoted to the properties of many-electron excitations of the 2D electron system in GaAs/AlGaAs structures in the integer and fractional quantum Hall regimes. Special attention is given to the study of the roton features in the spectra of fractional-gap excitations as well as to roton-like features in the Bernstein-branch spectra of quasiclassical magnetoplasmons. The properties of composite fermions are also investigated. Measurements of these excitation dispersions were conducted using a novel experimental technique that combines microwave spectroscopy with surface acoustic wave spectroscopy and an optical detection scheme.

This chapter summarizes hyperfine interactions in semiconductor quantum systems, especially those that are two-dimensional, with an emphasis on dynamical nuclear polarization and its detection based on the characteristics of quantum Hall ferromagnetism. Dynamic nuclear polarization (DNP) and its detection can be achieved for a single layer and even for a nanostructure, resulting in highly-sensitive resistivelydetected nuclear-magnetic-resonance (RD-NMR). RD-NMR enables us to extend powerful features of NMR to semiconductor quantum systems. Many interesting and important electron spin features have been revealed by sophisticated RD-NMR studies. Novel nuclear resonance mediated by electrically-induced domain oscillation is finally discussed as an approach toward microscopic nuclear manipulation in semiconductor quantum systems.

Recent studies of magnetotransport in high Landau levels of 2D electron systems (2DES) revealed a rich variety of nonlinear phenomena. The most prominent examples of these phenomena emerge when a 2DES is subject to external electric fields that drive the system out of equilibrium. In particular, when a 2DES is irradiated by monochromatic microwave radiation, its longitudinal magnetoresistivity exhibits prominent oscillations, known as microwave-induced resistance oscillations. At sufficiently high microwave power and low temperature, the minima of these oscillations can extend all the way to zero giving rise to zero-resistance states. This chapter reviews main experimental (and some theoretical) aspects of these phenomena.

In two closely spaced two-dimensional electron systems, inter-electron Coulomb forces can promote electron-hole pairing (exciton formation) and their Bose condensation at lowest temperatures. Even though an exciton is a charge neutral entity and should be indifferent to electric and magnetic fields, exciton condensates emerging in such two-dimensional electron systems display some remarkable magneto-transport phenomena and exhibit signatures of superfluidity. The following text provides an overview of some of the experiments which have been used to scrutinize this unique state of matter.

We overview the magnetotransport properties of the two-dimensional electron gas (2DEG) confined at the MgZnO/ZnO heterointerfaces. In this system, electrons are accumulated spontaneously at the interface due to the discontinuity between spontaneous polarization of MgZnO and that of ZnO. The electronmobility exceeds 700,000cm²V^-1s^-1 and one can observe integer and fractional quantum Hall (QH) effects. What is distinct from other 2DEGs in conventional semiconductors is the pronounced electron correlation effect due to the large electron effective mass in ZnO.We discuss the features of oxides in comparison with GaAs and suggest that the MgZnO/ZnO system can be a prototype system for highly correlated QH physics.

The following sections are included:

• Appendices
  • Energy Scales
  • Hausdorff Formulas
  • Group SU(2) and Pauli Matrices
  • Groups SU(N) and SU(2N)
  • Cauchy-Riemann Equations
  • Green Function
  • Bogoliubov Transformation
  • Energy-Momentum Tensor
  • Exchange Interaction
  • Mermin-Wagner Theorem
  • Lorentz Transformation
  • One-Dimensional Soliton Solutions
  • Field-Theoretical Vortex Operators
  • Bosonization in One-Dimensional Space
  • Coulomb Energy Formulas
  • W_∞ Algebra

• References

• Index