Electron Correlation in the Solid State

The following sections are included:

• Preface

• Contents

The following sections are included:

• Introduction
  • Ideal Fermi gas
  • Hartree–Fock energy
  • Screening and plasma oscillations
  • Plasmon contribution to the ground state energy and kinetic correlations
  • Wigner crystallization

• Dielectric Function
  • Dielectric function: definition and main general properties
  • Proper polarizability
  • Lindhard susceptibility and screening in the random phase approximation
  • Local field factor for exchange and correlation
  • Connections with density functional theory

• Some Applications and Recent Developments
  • Screened interactions in metals
  • Acoustic waves, transport coefficients and liquid structure in simple metals
  • Density wave theory of Wigner crystallization
  • Phonons in Wigner crystals near melting
  • Two-pair excitation spectrum at long wavelengths
  • Two-dimensional jellium and layered jellia

• References

The following sections are included:

• Introduction

• Weakly Correlated Systems
  • Projection techniques
  • Incremental method
  • Results for semiconductors and ionic crystals

• Strongly Correlated Electron Systems
  • Model Hamiltonians
  • Spectral densities
  • Application to 3d-transition metals
  • Spectral functions of Cu–O planes

• Electron Crystallization

• Heavy Fermions
  • Kondo lattices
  • Zeeman scenario – Nd_2−xCe_xCuO₄
  • Hubbard route: Yb₄As₃

• Acknowledgment

• References

In this contribution we deal with a number of theoretical aspects concerning physics of systems of interacting electrons. Our discussions, although amenable to appropriate generalisations, are subject to some limitations. To name, we deal with systems of spin-less fermions — or those of spin-compensated fermions with spin —, with nondegenerate ground states, and those in which relativistic effects are negligible; we disregard ionic motions and deal with “normal” (not superconducting, for instance) systems that are in addition free from randomly distributed impurities. We restrict our considerations to the absolute zero of temperature. The Green and response functions feature in our theoretical considerations. Here we give especial attention to the analytic properties of these functions for complex values of energy. We discuss how, both fundamentally and from the practical viewpoint, ground and low-lying excited-states properties can be obtained from these correlation functions. Characterising low-lying excited states by means of elementary excitations, we deal with both those that are particle-like (the Landau quasi-particles) and those that are collective (plasmons, excitation in the total distribution of electrons). We devote some space to discussions concerning the domain of validity and breakdown of the many-body perturbation theory, specifically that for the single-particle Green function and the self-energy operator. Extensive analysis of the asymptotic behaviour of dynamic correlation functions in the limits of small and large energies reveal the significance of the Kohn-Sham-like Hamiltonians within the context of the many-body perturbation theory. In view of this, at places we pay especial attention to a number of the existing density-functional theories (including the ones for the single-particle reduced density matrix and time-dependent external potentials). We discuss in some detail a number of issues that are specific to the (phenomenological) Landau Fermi-liquid theory and their justification within the framework of the many-body perturbation theory. In doing so we touch upon a number of characteristic features specific to Fermi-liquid (as oppsed to marginal Fermi- and Luttinger-liquid) systems. Finally, we put one particular approximation scheme for the self-energy operator, known as the the GW scheme, under magnifying glass and observe it in many of its facets.

The following sections are included:

• Outline and Introduction

• Definition of Fermi Liquid and Tomographic Luttinger Liquid States

• Anderson Anomaly in 2D

• Zero Sound and Failure of Fermi Liquid State

• Calculation

• Connection to Anderson's Proposal

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

• Tomographic Luttinger Liquid as an Ideal Gas of Condensed Fermionic Strings

• Zero Sound as RVB Gauge Fields

• Summary

• Acknowledgments

• References

The following sections are included:

• Introduction

• Scaling at Quantum Critical Points

• Fermionic Field Theory
  • Grassmannian field theory
  • Order parameter field theories
    • Magnetic order parameters
    • Superconducting order parameter
  • The nonlinear sigma-model
    • Digression: The nonlinear sigma-model for classical Heisenberg ferromagnets
    • Symmetry properties of the fermion model
    • Separation of soft and massive modes, and the nonlinear sigma-model for fermions
    • Order parameter field theory for metal-insulator transitions

• Magnetic Transitions at Zero Temperature
  • Itinerant ferromagnets
    • Disordered ferromagnets
    • Clean ferromagnets
  • Disordered antiferromagnets

• Superconductor–Metal Transition at Zero Temperature

• Metal–Insulator Transitions
  • Disordered Fermi liquids
    • The disordered Fermi liquid fixed point
    • Scaling behavior of observables
  • The Anderson–Mott transition
    • Anderson–Mott transition near two dimensions
    • Anderson–Mott transition in high dimensions
    • Anderson–Mott transitions in three dimensions: Conventional scaling scenario
    • Anderson–Mott transitions in three dimensions: Activated scaling scenario

• Acknowledgments

• References

The following sections are included:

• Introduction

• Density Matrices
  • Definitions and some properties
  • Natural orbitals
  • Van Hove correlation function

• Density Functional Theory: Exchange and Correlation Potentials
  • Differential virial theorems
  • Force-balance equations
  • Exact expression for the exchange-correlation potential, applicable to mixed-state systems
  • Exchange and correlation energy of mixed-state systems

• Density Matrices and Density Functionals in Strong Magnetic Fields
  • Outline and background
  • Orbital motion in magnetic field
    • The Hamiltonian
    • Equation of motion for the first-order density matrix in a magnetic field
    • Differential virial equation (DVE)
    • Interpretation of DVE as a force-balance equation
  • Inclusion of electron spin: relation to current density functional theory
    • Generalized virial
    • Current-density functional theory of Vignale and Rasolt
    • Approximate exchange-only potentials
    • Fractional quantum Hall liquid freezing into a Wigner solid
  • Summary of main results in magnetic field

• Time-Dependent Density Functional Theory
  • Calculation of excitation energies by time-dependent DFT
  • Response function theory
  • Exact equation for exchange–correlation potential in time-dependent density functional theory

• Illustrative Examples of Density Matrix and Density Functional Theory
  • Exchange potential at a jellium-type surface
  • Asymptotic properties of exchange-only potential v_x(r)
  • Density-matrix renormalization group study: one example
    • Interacting Fermions on a ring in presence of disorder
    • Friedel oscillations
  • Density functional theory (DFT) applied to positron states in solids
    • Positrons at vacancies

• Quantum Monte Carlo Studies: Brief Summary
  • Diffusion Monte Carlo studies
    • Importance sampling
  • Green function Monte Carlo approach
  • Monte Carlo computer experiments on phase transitions in uniform interacting electron assembly

• Acknowledgments

• Appendix 2.1. Some Further Properties of Density Matrices, Including Spin

• Appendix 4.1. Hartree-Fock Approximation in a Magnetic Field

• Appendix 4.2. Magnetic Field Dependent Density Functional and Density Matrix Theory; Form of Scalar Exchange-Correlation Potential
  • Exchange-correlation scalar potential

• References