Title Page

Indice

Preface

Gruppo fotografico dei partecipanti al Corso

QUANTUM MONTE CARLO STUDIES OF THE ELECTRON LIQUID

Introduction to Quantum Monte Carlo methods applied to the electron gas

Introduction

Variational Monte Carlo

Random walks

The variational method

The pair product trial function

Details

Optimization of trial functions

The spin-orbits: crystal vs. liquid

Beyond the pair product trial function

Finite-size effects

Problems with variational methods

Diffusion Monte Carlo

Importance sampling

The fixed-node method

Fixed-phase method

Exact fermion methods

Problems with projection methods

Ground-state properties of the electron gas in 2 and 3D

Fermi-liquid parameters

Polarization of the electron gas

Calculation of spin and charge response

Path integral Monte Carlo and applications

Defects in the 2D Wigner crystal

PIMC of exchange frequencies and magnetic ordering in the 2D Wigner crystal

DENSITY FUNCTIONAL THEORY

Electronic density functional theory: Ground-state properties

Introduction

Wave function approach to the ground-state problem

Electron number density

First Hohenberg-Kohn theorem: non-degenerate ground-state case

First Hohenberg-Kohn theorem: degenerate ground-state case

Second Hohenberg-Kohn theorem

Formal solution of the ground-state problem in the Hohenberg-Kohn form

Solution of the ground-state problem for a non-interacting system

Kohn-Sham approach to the ground-state density problem

Ground-state energy from the Kohn-Sham approach

Approximations for the exchange correlation energy functional

Density matrices

Spin density functional theory

Density functional theory in terms of orbital-dependent functionals

Hartree-Fock approximation to the ground-state problem

The role of the exact exchange energy and potential

Determination of the exact exchange potential

Dynamic correlations in the electron liquid

Introduction: the electron gas model in the normal state and its realizations

Structure: the electron pair distribution functions

The exchange hole

The exchange-correlation hole

Relation to diffractive and inelastic scattering functions

Relation to screening

Approximate theories of exchange-correlation holes

The STLS approach

The Overhauser scattering approach

The Fermi hypernetted-chain approach

Spin ordering in the electron gas

Dielectric response and spin response

Definitions and general properties of density response

Free electrons and random phase approximation

Spin-resolved response

Many-body local-field factors

LFF at large wave number

Thermodynamic limit

Spin waves and spin diffusion

Effective interactions in the electron gas

Correlation energy from scaled exchange-correlation kernels

Dynamic correlations

Plasmons and multi-pair excitations

Current-density functional theory

Generalized hydrodynamics

Current response functions

Asymptotic behaviours of the local field factors

Two-pair excitation spectra

Relation to Boltzmann equation approach

Appendix A. Magnetic virial theorem

The electron gas in TDDFT and SCDFT

Introduction

The electron gas paradigm

Ground-state density functional theory

Ground-state exchange-correlation functionals

Time-dependent density functional theory

Preliminaries

Basic theorems

The Runge-Gross theorem

Causality and the quantum-mechanical action

Time-dependent Kohn-Sham equations

Time-dependent exchange-correlation potentials

Linear response theory

The response function

The poles of the response function

Approximations to the exchange-correlation kernel

Some results for finite systems

Some results for extended systems

The exchange-correlation kernel of the homogeneous electron gas

Exact features of f unif xc

Approximations to f unif xc

Density functional theory for superconductors

Preliminaries

Basic theorems

An LDA for superconductors

The local spin density approximation

Construction of an LDA for superconductors

How to calculate epsilon unif xc

Construction of the explicit functional

Conclusions

Conserving approximations in nonequilibrium Green function and density functional theory

Introduction

Nonequilibrium Green function theory

The action functional

The Kadanoff-Baym equations

Conserving approximations

Momentum conservation

Time-dependent density functional theory

The Sham-Schluter equation

Summary

Appendix A. The frequency sum rule

ELECTRONS IN SEMICONDUCTORS

Spin transport and quantum magnetism in semiconductors

Introduction

Lecture 1: Spin-dependent transport in "magnetic" two-dimensional electron gases

Lecture 2: Ferromagnetic semiconductors

Lecture 3: Optical studies of coherent spin transport in semiconductors

Coulomb drag effect in coupled electron systems

Introduction

Theory

Model of a bilayer electron system

Drag resistivity

Effective inter-layer interaction

Results

Conclusions and future prospects

The classical electron gas in artificial structures

Introduction

Artificial atoms

General Hamiltonian and the classical limit

Experimental realizations

The numerical approach

Classical artificial atoms

Normal modes

Melting

Coupled quantum dots: artificial molecules

Model system

Small molecules

Larger molecules

METAL-INSULATOR TRANSITION AND FERMI-LIQUID PROPERTIES

Disordered electron systems

Introduction

Setting the stage for the metal-insulator transition

The semiclassical approach of Drude-Boltzmann

The metal-insulator transition

The Anderson transition and quantum interference

The scaling theory of the metal-insulator transition

The microscopic approach

Linear response theory, Kubo formula and all that

Conservation laws and gauge invariance

Response functions and Ward identities

Non-interacting disordered electrons

Self-consistent Born approximation

Vertex part and diffuson ladder

Weak localization

Effect of a magnetic field

Spin effects

A review of the experimental situation

Interacting disordered electrons

Perturbation theory and the search for the effective couplings

Density of states

Electrical conductivity

Thermodynamic potential

The renormalized perturbation theory and effective Fermi-liquid description

Effective scattering amplitudes and Landau parameters

Renormalized response functions

Derivation of the group equations

The renormalization group equations

Appendix A. Useful integrals

Appendix B. Magnetic impurities

Appendix C. Spin-orbit scattering

Appendix D. The long-range interaction case

Appendix E. Details on the evaluation of the interaction correction to the conductivity

Appendix F. Details of the evaluation of the thermodynamic potential

Appendix G. Ladder in the presence of Zeeman coupling

Appendix H. The ladder renormalization

Metallic conduction, apparent metal-insulator transition and related phenomena in two-dimensional electron liquid

Introduction

Mott semiclassical picture of the MIT

Basics of the 2D semiconducting devices

Quantum transport at zero field

Various types of transport: Delocalized states: diffusive and ballistic transport

Localized states

Electron's phase coherence and transport

Suppression of the weak localization in H fields

Single-particle scaling theory of localization

Spin-orbit scattering case

Interaction quantum corrections in the diffusive regime

Interaction quantum corrections to the transport in the ballistic regime

An apparent MIT in 2D

Quantitative studies of the electron-electron interactions

Fermi-liquid renormalization of electron parameters in 2D systems

Implementation of the measured FL parameters to the metallic-like transport

Diffusive regime, T tau << 1

Ballistic regime, T tau > 1

Comparison of the theory with experiment at high G >> e 2/h

Transport in the critical regime, sigma ~ e 2/h and n ~= n c

Transport in the presence of the in-plane field

Regime of high densities n >> n c, G >> e 2/h

Critical regime of densities n ~= nc, and sigma ~= e 2/h

Homogeneity

Summary

A brief guide to electronic lifetimes in 2D

Introduction

Outline

Scattering mechanisms in 2D

Lifetime definitions

Lifetime measurements

2D-2D tunneling spectroscopy

Comparison of experimental methods

Electron-electron scattering

Conclusions

Determining spin susceptibility in a variable-density two-dimensional electron system using two methods

Introduction

Measurement techniques and results

Tilted-field method

Parallel-field method

Discussions

Conclusion

HEAVY FERMIONS AND LATTICE MODELS

Aspects of magnetism and superconductivity in metals

Magnetic moments and electrons in metals

Introduction

Heavy (slow) electrons

Non-Fermi-liquid features

Unconventional superconductivity of heavy-electron metals

Magnetism and its influence on electrons near metal-insulator transitions

Cu oxides

The normal state

Unconventional superconductivity of cuprates

Mn oxides

Hexaborides

Electronic and magnetic properties of EuB 6

Magnetotransport in Eu 1-x Ca x B 6

Summary

Instabilities of the 2-dimensional electron liquid

Introduction

Simple introduction to RG methods

One-dimensional systems

Single chain

Two-leg ladder

The two-dimensional t-t'-U model

Introduction

RG theory in two dimensions

RG flow for the case of electron doping

RG flows at the van Hove filling

The crossover between AF and d-wave pairing

Conclusions

Metals with heavy quasiparticles

General features

Kondo lattices: renormalized band structures

Partially localized 5f electrons

Charge ordering

The case of d electrons

QUANTUM HALL SYSTEMS

Composite fermions in quantum Hall systems

Introduction

Electrons confined to a two-dimensional surface in a perpendicular magnetic field

Integer quantum Hall effect

Fractional quantum Hall effect

Numerical studies

Chern-Simons gauge field

Jain's composite fermion picture

Pseudopotentials

Angular momentum

Coefficients of fractional parentage

Non-harmonic pseudopotentials and correlations

Correlations in higher Landau levels

Chern-Simons gauge field revisited

Gedanken experiment: Laughlin states and the Jain sequence

The composite fermion hierarchy

Quasiparticle-quasiparticle interactions

Quasiparticle-quasiparticle pairing and novel families of incompressible states

Composite fermions: Edge states and fractional statistics

Introduction

Edge Luttinger liquid

Equal-time edge Green function

Corrections due to CF quasi-LL mixing

Comparison with experiment

Fractional statistics

Fractional statistics in the composite fermion theory

Microscopic confirmation

Discussion

Electronic correlations at the edge of a quantum Hall liquid

Summary

Introduction to quantum Hall edges

Edge tunneling experiment

Edge tunneling results and analysis

Exponent at Laughlin fractions: 1/nu = 1,3

Exponent over continuum range 1 < 1/nu < 4

Open questions

Outlook

Conclusion

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