Electron Correlation in the Solid State

by March, Norman H.

• ISBN13: 9781860942006
• ISBN10: 1860942008
• Format: Hardcover
• Copyright: 1999-08-01
• Publisher: Imperial College Pr

Summary

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.

Table of Contents

Preface	v
Many-Body Effects in Jellium	1	(46)
M. P. Tosi
Introduction	1	(7)
Ideal Fermi gas	2	(1)
Hartree-Fock energy	2	(2)
Screening and plasma oscillations	4	(1)
Plasmon contribution to the ground state energy and kinetic correlations	5	(1)
Wigner crystallization	6	(2)
Dielectric Function	8	(17)
Definition and main general properties	8	(4)
Proper polarizability	12	(1)
Lindhard susceptibility and screening in the random phase approximation	13	(6)
Local field factor for exchange and correlation	19	(5)
Connections with density functional theory	24	(1)
Some Applications and Recent Developments	25	(22)
Screened interactions in metals	25	(3)
Acoustic waves, transport coefficients and liquid structure in simple metals	28	(4)
Density wave theory of Wigner crystallization	32	(1)
Phonons in Wigner crystals near melting	33	(2)
Two-pair excitation spectrum at long wavelengths	35	(5)
Two-dimensional jellium and layered jellia	40	(2)
References	42	(5)
Solids with Weak and Strong Electron Correlations	47	(56)
P. Fulde
Introduction	47	(3)
Weakly Correlated Systems	50	(16)
Projection techniques	51	(3)
Incremental method	54	(1)
Results for semiconductors and ionic crystals	55	(11)
Strongly Correlated Electron Systems	66	(13)
Model Hamiltonians	67	(3)
Spectral densities	70	(2)
Application to 3d-transition metals	72	(4)
Spectral functions of Cu--O planes	76	(3)
Electron Crystallization	79	(4)
Heavy Fermions	83	(20)
Kondo lattices	85	(5)
Zeeman scenario -- Nd2-xCexCuO4	90	(5)
Hubbard route: Yb4As3	95	(2)
Acknowledgment	97	(1)
References	97	(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