Electron Correlation in the Solid State

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ISBN: 9781860942006 | 1860942008
Cover: Hardcover
Copyright: 8/1/1999

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Treats the many-electron theory of the solid state. Suitable for research workers and higher degree students in a number of disciplines: theoretical physics, materials science, solid-state chemistry all being embraced.

Preface

Many-Body Effects in Jellium

(46)

M. P. Tosi

Introduction

(7)

Ideal Fermi gas

(1)

Hartree-Fock energy

(2)

Screening and plasma oscillations

(1)

Plasmon contribution to the ground state energy and kinetic correlations

(1)

Wigner crystallization

(2)

Dielectric Function

(17)

Definition and main general properties

(4)

Proper polarizability

(1)

Lindhard susceptibility and screening in the random phase approximation

(6)

Local field factor for exchange and correlation

(5)

Connections with density functional theory

(1)

Some Applications and Recent Developments

(22)

Screened interactions in metals

(3)

Acoustic waves, transport coefficients and liquid structure in simple metals

(4)

Density wave theory of Wigner crystallization

(1)

Phonons in Wigner crystals near melting

(2)

Two-pair excitation spectrum at long wavelengths

(5)

Two-dimensional jellium and layered jellia

(2)

References

(5)

Solids with Weak and Strong Electron Correlations

(56)

P. Fulde

Introduction

(3)

Weakly Correlated Systems

(16)

Projection techniques

(3)

Incremental method

(1)

Results for semiconductors and ionic crystals

(11)

Strongly Correlated Electron Systems

(13)

Model Hamiltonians

(3)

Spectral densities

(2)

Application to 3d-transition metals

(4)

Spectral functions of Cu--O planes

(3)

Electron Crystallization

(4)

Heavy Fermions

(20)

Kondo lattices

(5)

Zeeman scenario -- Nd2-xCexCuO4

(5)

Hubbard route: Yb4As3

(2)

Acknowledgment

(1)

References

(6)

Ground and Low-Lying Excited States of Interacting Electron Systems; a Survey and Some Critical Analyses

103

(160)

B. Farid

Introduction

104

(9)

Mathematical Preliminaries

113

(11)

Types of singularity

113

(1)

Many-valued functions: Physical and non-physical Riemann sheets

114

(3)

Series and asymptotic series

117

(3)

Physical motivation

120

(4)

Generalities

124

(1)

The Single-Particle Green Function

125

(15)

The Lehmann representation for G(ε)

126

(1)

On the ``chemical potential'' μ

127

(2)

Sums involving the Lehmann amplitudes and energies

129

(3)

A symmetry property of G

132

(2)

Analytic continuation of G(ε), G(z)

134

(2)

Large-|ε| behavior of G(ε)

136

(1)

G(z) is invertible

137

(1)

Connection between analytic continuation and choice of representation

138

(2)

The (Proper) Self-Energy Σ(ε) and its Analytic Continuation Σ(ε)

140

(5)

Analyticity of Σ(ε) and some consequences

140

(4)

A ``local-density'' approximation for Σ(ε)

144

(1)

Quasi-Particles; Particle-Like Excitations

145

(25)

The quasi-particle approximation

150

(3)

Quasi-particle energies: Poles and non-isolated singularities

153

(1)

Quasi-particles in homogeneous systems

154

(4)

Fermi-versus non-Fermi liquid; a Luttinger's theorem revisited

158

(4)

Breakdown of the many-body perturbation theory?

162

(3)

Momentum-distribution function; a Migdal's theorem revisited

165

(2)

Some comments concerning solutions of the quasi-particle equation

167

(3)

Determination of the Single-Particle Green Function

170

(14)

Exact approach

170

(1)

Equation-of-motion approach and truncation of hierarchy

170

(1)

Conserving approximations

171

(1)

Many-body perturbation theory and its breakdown

172

(1)

In defence of the many-body perturbation theory

172

(5)

Dyson's argument

177

(1)

Simon's argument; two counter examples

177

(2)

Set of self-sufficient equations

179

(3)

Two functional forms for the self-energy operator, Σ{0} and Σ{1}

182

(2)

The Density-Density Correlation Function χ and the Polarisation Function P

184

(26)

Symmetries of χ and P

185

(1)

Analytic continuation of χ(ε), χ(ε); analyticity and its consequences

186

(2)

Large-|ε| limits

188

(2)

Perturbation expansion for the polarisation function P; P{0} and P{1}

190

(1)

Random-Phase Approximation, RPA, and large |ε|

191

(2)

On aspects of the density-functional theory

193

(1)

The ground-state DFT

194

(2)

The time-dependent DFT

196

(1)

The local-field function G

197

(1)

Quasi-particles; collective charge excitations (Plasmons)

198

(5)

Pair-correlation functions

203

(2)

The continued-fraction expansion and its physical significance

205

(3)

A plasmon-pole approximation

208

(2)

The GW Approximation for the Self-Energy

210

(30)

Some historical background

210

(1)

Details of the GW approximation exposed (Part I)

211

(4)

Some sum-rules concerning ΣGW

215

(1)

Details of the GW approximation exposed (Part II)

216

(3)

Some approximate schemes within the GW approximation

219

(1)

Large-|ε| behaviour of ΣGW(ε)

220

(1)

The DFT revisited: An explicitly non-local effective potential

221

(4)

Self-consistent calculations

225

(3)

Some technical aspects

228

(4)

A survey of computational works within the GW approximation

232

(4)

Simplified schemes and suggestions

236

(2)

General trends and vertex corrections

238

(2)

Summary and Concluding Remarks

240

(23)

Appendix A: On the Representation Spaces and Some Conventions

248

(1)

Appendix B: Discontinuity in the Time Domain versus Asymptotic Behaviour in the Energy Domain

249

(1)

Acknowledgments

250

(1)

List of Some Symbols

251

(1)

List of Abbreviations and Acronyms

252

(1)

References and Literature

253

(10)

Failure of Fermi Liquid Theory in Two and Three Dimensions

263

(34)

G. Baskaran

Outline and Introduction

263

(2)

Definition of Fermi Liquid and Tomographic Luttinger Liquid States

265

(2)

Anderson Anomaly in 2D

267

(9)

Zero Sound and Failure of Fermi Liquid State

276

(1)

Calculation

277

(11)

Connection to Anderson's Proposal

288

(1)

RG Analysis and Failure of Fermi Liquid Theory in Two and Three Dimensions

289

(3)

Tomographic Luttinger Liquid as an Ideal Gas of Condensed Fermionic Strings

292

(1)

Zero Sounds as RVB Gauge Fields

293

(1)

Summary

294

(3)

Acknowledgments

294

(1)

References

295

(2)

Quantum Phase Transitions in Electronic Systems

297

(74)

T. R. Kirkpatrick

D. Belitz

Introduction

297

(7)

Scaling at Quantum Critical Points

304

(5)

Fermionic Field Theory

309

(19)

Grassmannian field theory

309

(4)

Order parameter field theories

313

(1)

Magnetic order parameters

313

(2)

Superconducting order parameter

315

(1)

The nonlinear sigma-model

316

(1)

Digression: The nonlinear sigma-model for classical Heisenberg ferromagnets

316

(4)

Symmetry properties of the fermion model

320

(4)

Separation of soft and massive modes, and the nonlinear sigma-model for fermions

324

(2)

Order parameter field theory for metal-insulator transitions

326

(2)

Magnetic Transitions at Zero Temperature

328

(13)

Itinerant ferromagnets

328

(1)

Disordered ferromagnets

329

(5)

Clean ferromagnets

334

(3)

Disordered antiferromagnets

337

(4)

Superconductor--Metal Transition at Zero Temperature

341

(4)

Metal--Insulator Transitions

345

(26)

Disordered Fermi liquids

345

(1)

The disordered Fermi liquid fixed point

346

(2)

Scaling behavior of observables

348

(2)

The Anderson--Mott transition

350

(1)

Anderson--Mott transition near two dimensions

351

(6)

Anderson--Mott transition in high dimensions

357

(3)

Anderson--Mott transition in three dimensions: Conventional scaling scenario

360

(3)

Anderson--Mott transition in three dimensions: Activated scaling scenario

363

(3)

Acknowledgments

366

(1)

References

366

(5)

Density Matrices, Density Functional Theory and Quantum Monte Carlo Calculations

371

(52)

N. H. March

Introduction

371

(1)

Density Matrices

372

(4)

Definitions and some properties

372

(1)

Natural orbitals

373

(3)

Van Hove correlation function

376

(1)

Density Functional Theory: Exchange and Correlation Potentials

376

(10)

Differential virial theorems

377

(2)

Force-balance equations

379

(1)

Exact expression for the exchange-correlation potential, applicable to mixed-state systems

380

(3)

Exchange and correlation energy of mixed state systems

383

(3)

Density Matrices and Density Functionals in Strong Magnetic Fields

386

(17)

Outline and background

386

(2)

Orbital motion in magnetic field

388

(1)

The Hamiltonian

388

(1)

Equation of motion for the first-order density matrix in a magnetic field

389

(1)

Differential virial equation (DVE)

390

(2)

Interpretation of DVE as a force-balance equation

392

(2)

Inclusion of electron spin: relation to current density functional theory

394

(1)

Generalized virial

394

(2)

Current--density functional theory of Vingnale and Rasolt

396

(2)

Approximate exchange-only potentials

398

(2)

Fractional quantum Hall liquid freezing into a Wigner solid

400

(2)

Summary of main results in magnetic field

402

(1)

Time-Dependent Density Functional Theory

403

(5)

Calculation of excitation energies by time-dependent DFT

404

(1)

Response function theory

405

(2)

Exact equation for exchange-correlation potential in time-dependent density functional theory

407

(1)

Illustrative Examples of Density Matrix and Density Functional Theory

408

(7)

Exchange potential at a jellium-type surface

409

(1)

Asymptotic properties of exchange-only potential vx(r)

410

(1)

Density matrix renormalization group study: one example

411

(1)

Interacting Fermions on a ring in presence of disorder

411

(1)

Friedel oscillations

411

(2)

Density functional theory (DFT) applied to positron states in solids

413

(1)

Positrons at vacancies

414

(1)

Quantum Monte Carlo Studies: Brief Summary

415

(8)

Diffusion Monte Carlo studies

416

(1)

Importance sampling

417

(2)

Green function Monte Carlo approach

419

(2)

Monte Carlo computer experiments on phase transitions in uniform interacting electron assembly

421

(1)

Acknowledgments

422

(1)

Appendices

423

(9)

A2.1 Some further properties of density matrices, including spin

423

(3)

A4.1 Hartree--Fock approximation in a magnetic field

426

(2)

A4.2 Magnetic field dependent density functional and density matrix theory; forms of exchange--correlation potential

428

(4)

4.2.1 Exchange--correlation scalar potential

428

(4)

References

432

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Electron Correlation in the Solid State

Summary

Table of Contents

Supplemental Materials