interacting electrons theory and computational...

INTERACTING ELECTRONS

Theory and Computational Approaches

Recent progress in the theory and computation of electronic structure is bringing anunprecedented level of capability for research. Many-body methods are becoming essentialtools vital for quantitative calculations and understanding materials phenomena in physics,chemistry, materials science, and other fields. This book provides a unified exposition ofthe most-used tools: many-body perturbation theory, dynamical mean-field theory, andquantum Monte Carlo simulations.

Each topic is introduced with a less technical overview for a broad readership, followedby in-depth descriptions and mathematical formulation. Practical guidelines, illustrations,and exercises are chosen to enable readers to appreciate the complementary approaches,their relationships, and the advantages and disadvantages of each method. This book isdesigned for graduate students and researchers who want to use and understand theseadvanced computational tools, get a broad overview, and acquire a basis for participatingin new developments.

RICHARD M. MARTIN is Emeritus Professor at the University of Illinois at Urbana-Champaign and Consulting Professor at Stanford University. He has made extensivecontributions to the field of modern electronic structure methods and the theory of inter-acting electron systems and he is the author of the companion book, Electronic Structure:Basic Theory and Methods.

LUCIA REINING is CNRS Senior Researcher at École Polytechnique Palaiseau andfounding member of the European Theoretical Spectroscopy Facility. Her work coversmany-body perturbation theory and time-dependent density functional theory. She is arecipient of the CNRS silver medal and fellow of the American Physical Society.

DAVID M. CEPERLEY is a Founders and Blue Waters Professor at the University of Illi-nois at Urbana-Champaign where he has pioneered the quantum Monte Carlo method,including the development of variational, diffusion, and path-integral Monte Carlo. He isa member of the US National Academy of Sciences and recipient of the Rahman Prize forComputational Physics of the APS and the Feenberg Medal for Many-Body Physics.

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Cambridge University Press978-0-521-87150-1 - Interacting Electrons: Theory and Computational ApproachesRichard M. Martin, Lucia Reining and David M. CeperleyFrontmatterMore information

http://www.cambridge.org/9780521871501

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Interacting ElectronsTheory and Computational Approaches

Richard M. MartinUniversity of Illinois Urbana-Champaign

Lucia ReiningÉcole Polytechnique Palaiseau

David M. CeperleyUniversity of Illinois Urbana-Champaign






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c© Cambridge University Press 2016

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First published 2016

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Library of Congress Cataloguing-in-Publication DataNames: Martin, Richard M., 1942– author. | Reining, Lucia, author. | Ceperley, David, author.

Title: Interacting electrons : theory and computational approaches / Richard M. Martin(University of Illinois, Urbana-Champaign), Lucia Reining (Ecole Polytechnique, Palaiseau),

David M. Ceperley (University of Illinois, Urbana-Champaign).Description: New York, NY : Cambridge University Press, [2016] | ?2016

Identifiers: LCCN 2015041121 | ISBN 9780521871501 (hardback) |ISBN 0521871506 (hardback)

Subjects: LCSH: Electronic structure. | Electrons. | Many-body problem. |Perturbation (Quantum dynamics) | Quantum theory. | Monte Carlo method.

Classification: LCC QC176.8.E4 M368 2016 | DDC 539.7/2112–dc23LC record available at http://lccn.loc.gov/2015041121

ISBN 978–0–521-87150-1 Hardback

Cambridge University Press has no responsibility for the persistence or accuracyof URLs for external or third-party internet websites referred to in this publication,

and does not guarantee that any content on such websites is, or will remain,accurate or appropriate.






To our families






Contents

Preface page xvii

Acknowledgments xix

Notation xx

Part I Interacting electrons: beyond the independent-particle picture

1 The many-electron problem: introduction 1Summary 1

1.1 The electronic structure problem 21.2 Why is this problem hard? 31.3 Why is the independent-electron picture so successful? 51.4 Development of theoretical approaches to the many-body problem 81.5 The many-body problem and computation 101.6 The scope of this book 13

Select further reading 14

2 Signatures of electron correlation 15Summary 15

2.1 What is meant by correlation? 162.2 Ground-state and thermodynamic properties 172.3 Magnetism and local moments 202.4 Electron addition and removal: the bandgap problem and more 212.5 Satellites and sidebands 262.6 Particle–hole and collective excitations 292.7 The Kondo effect and heavy fermions 322.8 Mott insulators and metal–insulator transitions 332.9 Lower dimensions: stronger interaction effects 362.10 Wrap-up 39

3 Concepts and models for interacting electrons 40Summary 40

3.1 The Wigner transition and the homogeneous electron system 403.2 The Mott transition and the Hubbard model 433.3 Magnetism and spin models 48






viii Contents

3.4 Normal metals and Fermi liquid theory 493.5 The Kondo effect and the Anderson impurity model 513.6 The Luttinger theorem and the Friedel sum rule 54

Select further reading 56Exercises 56

Part II Foundations of theory for many-body systems

4 Mean fields and auxiliary systems 59Summary 59

4.1 The Hartree and Hartree–Fock approximations 614.2 Weiss mean field and the Curie–Weiss approximation 644.3 Density functional theory and the Kohn–Sham auxiliary system 654.4 The Kohn–Sham electronic structure 704.5 Extensions of the Kohn–Sham approach 724.6 Time-dependent density and current density functional theory 764.7 Symmetry breaking in mean-field approximations and beyond 784.8 Wrap-up 80


5 Correlation functions 84Summary 84

5.1 Expectation values and correlation functions 855.2 Static one-electron properties 865.3 Static two-particle correlations: density correlations and the structure

factor 905.4 Dynamic correlation functions 935.5 Response functions 985.6 The one-particle Green’s function 1055.7 Useful quantities derived from the one-particle Green’s function 1115.8 Two-particle Green’s functions 116


6 Many-body wavefunctions 122Summary 122

6.1 Properties of the many-body wavefunction 1236.2 Boundary conditions 1246.3 The ground-state wavefunction of insulators 1266.4 Correlation in two-electron systems 1296.5 Trial function local energy, Feynman–Kac formula, and wavefunction

quality 1316.6 The pair product or Slater–Jastrow wavefunction 134






Contents ix

6.7 Beyond Slater determinants 139Exercises 141

7 Particles and quasi-particles 144Summary 144

7.1 Dynamical equations and Green’s functions for coupled systems 1457.2 The self-energy and the Dyson equation 1487.3 Illustration: a single state coupled to a continuum 1517.4 Interacting systems: the self-energy and spectral function 1527.5 Quasi-particles 1577.6 Quasi-particle equations 1617.7 Separating different contributions to a Dyson equation 1637.8 Wrap-up 165


8 Functionals in many-particle physics 169Summary 169

8.1 Density functional theory and the Hartree–Fock approximation 1718.2 Functionals of the Green’s function G and self-energy � 1748.3 Functionals of the screened interaction W 1798.4 Generating functionals 1828.5 Conservation laws and conserving approximations 1878.6 Wrap-up 190


Part III Many-body Green’s function methods

9 Many-body perturbation theory: expansion in the interaction 193Summary 193

9.1 The Coulomb interaction and perturbation theory 1949.2 Connecting the interacting and non-interacting systems 1999.3 Telling the story of particles: diagrams 2029.4 Making the story easier: two theorems 2069.5 Dyson equation for the one-particle Green’s function, and the

self-energy 2129.6 Diagrammatic expansion at non-vanishing temperature 2139.7 Self-consistent perturbation theory: from bare to dressed building

blocks 2159.8 The Luttinger–Ward functional 2179.9 Wrap-up 219







x Contents

10 Many-body perturbation theory via functional derivatives 222Summary 222

10.1 The equation of motion 22310.2 The functional derivative approach 22610.3 Dyson equations 22810.4 Conservation laws 23110.5 A starting point for approximations 23410.6 Wrap-up 242


11 The RPA and the GW approximation for the self-energy 245Summary 245

11.1 Hedin’s equations 24611.2 Neglecting vertex corrections in the polarizability: the RPA 25111.3 Neglecting vertex corrections in the self-energy: the GW approximation 25311.4 Link between the GWA and static mean-field approaches 26011.5 Ground-state properties from the GWA 26211.6 The GWA in the homogeneous electron gas 26511.7 The GWA in small model systems 27211.8 Wrap-up 277


12 GWA calculations in practice 280Summary 280

12.1 The task: a summary 28112.2 Frequently used approximations 28312.3 Core and valence 28912.4 Different levels of self-consistency 29212.5 Frequency integrations 29812.6 GW calculations in a basis 30212.7 Scaling and convergence 30612.8 Wrap-up 308


13 GWA calculations: illustrative results 311Summary 311

13.1 From the HEG to a real semiconductor: silicon as a prototype system 31213.2 Materials properties in the GWA: an overview 31913.3 Energy levels in finite and low-dimensional systems 32613.4 Transition metals and their oxides 32913.5 GW results for the ground state 337






Contents xi

13.6 A comment on temperature 34113.7 Wrap-up 343


14 RPA and beyond: the Bethe–Salpeter equation 345Summary 345

14.1 The two-particle correlation function and measurable quantities 34614.2 The two-particle correlation function: basic relations 34814.3 The RPA: what can it yield? 35014.4 Beyond the RPA: spin and frequency structure of the BSE 35314.5 The Bethe–Salpeter equation in the GW approximation 35514.6 A two-body Schrödinger equation 35714.7 Importance and analysis of electron–hole interaction effects 36114.8 Bethe–Salpeter calculations in practice 36814.9 Applications 37214.10 Extensions 37914.11 Linear response using Green’s functions or density functionals 38214.12 Wrap-up 385


15 Beyond the GW approximation 389Summary 389

15.1 The need to go beyond GW: analysis and observations 39115.2 Iterating Hedin’s equations 39315.3 Effects of vertex corrections 39415.4 The T-matrix and related approximations 39915.5 Beyond the T-matrix approximation: combining channels 40215.6 T-matrix and related approaches in practice 40615.7 Cumulants in electron spectroscopy 41015.8 Use of exact constraints 41515.9 Retrospective and outlook 417


16 Dynamical mean-field theory 421Summary 421

16.1 Auxiliary systems and embedding in Green’s function methods 42316.2 Overview of DMFT 42516.3 Expansion around an atomic limit: low energy scales and strong

temperature dependence 42916.4 Background for mean-field theories and auxiliary systems 43116.5 Dynamical mean-field equations 435






xii Contents

16.6 Self-energy functional and variational equations 44116.7 Static properties and density matrix embedding 44216.8 Single-site DMFA in a two-site model 44416.9 The Mott transition in infinite dimensions 44516.10 Hybridized bands and consequences for the Mott transition 45016.11 Interacting bands and spin transitions 45116.12 Wrap-up 453


17 Beyond the single-site approximation in DMFT 457Summary 457

17.1 Supercells and clusters 45817.2 Cellular DMFA 46017.3 Dynamic cluster approximation 46317.4 Variational cluster and nested cluster approximations 46617.5 Extended DMFT and auxiliary bosons 46717.6 Results for Hubbard models in one, two, and three dimensions 47017.7 Wrap-up 475


18 Solvers for embedded systems 479Summary 479

18.1 The problem(s) to be solved 48018.2 Exact diagonalization and related methods 48118.3 Path-integral formulation in terms of the action 48318.4 Auxiliary-field methods and the Hirsch–Fye algorithm 48518.5 CTQMC: expansion in the interaction 48718.6 CTQMC: expansion in the hybridization 49118.7 Dynamical interactions in CTQMC 49618.8 Other methods 49818.9 Wrap-up 499


19 Characteristic hamiltonians for solids with d and f states 502Summary 502

19.1 Transition elements: atomic-like behavior and local moments 50319.2 Hamiltonian in a localized basis: crystal fields, bands, Mott–Hubbard

vs. charge transfer 50719.3 Effective interaction hamiltonian 51219.4 Identification of localized orbitals 513






Contents xiii

19.5 Combining DMFT and DFT 51519.6 Static mean-field approximations: DFT+U, etc. 52219.7 Wrap-up 525


20 Examples of calculations for solids with d and f states 527Summary 527

20.1 Kondo effect in realistic multi-orbital problems 52820.2 Lanthanides – magnetism, volume collapse, heavy fermions, mixed

valence, etc. 52920.3 Actinides – transition from band to localized 53620.4 Transition metals – local moments and ferromagnetism: Fe and Ni 53720.5 Transition metal oxides: overview 54020.6 Vanadium compounds and metal–insulator transitions 54120.7 NiO – charge-transfer insulator, antiferromagnetism, and doping 54320.8 MnO – metal–insulator and spin transitions 54720.9 Wrap-up 549


21 Combining Green’s functions approaches: an outlook 553Summary 553

21.1 Taking advantage of different Green’s function methods 55521.2 Partitioning the system 55721.3 Combining different levels of diagrammatic approaches 55921.4 Combining Green’s function methods: GW and DMFT 56121.5 Dynamical interactions and constrained RPA 56821.6 Consequences of dynamical interactions 57021.7 Diagrammatic extensions: dynamical vertex approximation and dual

fermions 57121.8 Wrap-up 574


Part IV Stochastic methods

22 Introduction to stochastic methods 577Summary 577

22.1 Simulations 57822.2 Random walks and Markov chains 57922.3 The Metropolis Monte Carlo method 58122.4 Computing error bars 583






xiv Contents

22.5 The “heat bath” algorithm 58622.6 Remarks 587


23 Variational Monte Carlo 590Summary 590

23.1 Details of the variational Monte Carlo method 59223.2 Optimizing trial wavefunctions 59623.3 The momentum distribution and single-particle density matrix 59823.4 Non-local pseudopotentials 59923.5 Finite-size effects 60123.6 VMC for lattice models 60323.7 Excitations and orthogonality 60323.8 Strengths and weaknesses of VMC 606


24 Projector quantum Monte Carlo 609Summary 609

24.1 Types and properties of projectors 61024.2 The diffusion Monte Carlo method 61224.3 Exact fermion methods: the sign or phase problem 62124.4 The fixed-node and fixed-phase methods 62324.5 Mixed estimators, exact estimators, and the overlap 62824.6 Non-local pseudopotentials in PMC 63024.7 Projector auxiliary-field quantum Monte Carlo methods 63224.8 Applications of projector MC 63624.9 The pluses and minuses of projector MC 639


25 Path-integral Monte Carlo 644Summary 644

25.1 The path-integral representation 64525.2 Exchange of localized electrons 65025.3 Quantum statistics and PIMC 65225.4 Ground-state path integrals (GSPI) 65925.5 Finite-temperature QMC for the Hubbard model 66225.6 Estimating real-time correlation functions 66525.7 Correlation-function QMC for excitations 669


26 Concluding remarks 674






Contents xv

Part V Appendices

Appendix A Second quantization 677Summary 677

A.1 First quantization 677A.2 Second quantization 678


Appendix B Pictures 683Summary 683

B.1 Schrödinger picture 684B.2 Heisenberg picture 684B.3 Interaction picture 686


Appendix C Green’s functions: general properties 690Summary 690

C.1 Green’s functions for differential equations 690C.2 Fourier transforms and spectral representations 691C.3 Frequency integrals 693C.4 From many-body to few-body Green’s functions 695C.5 The thermodynamic limit 696


Appendix D Matsubara formulation for Green’s functions for T �= 0 699Summary 699

D.1 Green’s functions at T �= 0: Matsubara frequencies 699D.2 Analytic properties in the complex-frequency plane 702D.3 Illustration of the structure of G0(iωn) and G0(τ ) 705D.4 The grand potential � 707D.5 Transformation to real frequencies 709


Appendix E Time ordering, contours, and non-equilibrium 710Summary 710

E.1 The task 710E.2 The contour interpretation 710E.3 Contours for all purposes 712


Appendix F Hedin’s equations in a basis 715Summary 715

F.1 Generalization of Hedin’s equations 715






xvi Contents

F.2 Hedin’s equations in a basis 717Select further reading 717

Appendix G Unique solutions in Green’s function theory 719Summary 719

G.1 Which G0? Boundary conditions in time 719G.2 Which G? Self-consistent Dyson equations 720G.3 Convergence of perturbation expansions and consequences 721


Appendix H Properties of functionals 724Summary 724

H.1 Functionals and functional equations 724H.2 Legendre transformations and invertibility 725H.3 Examples of functionals for the total energy in Kohn–Sham DFT

calculations 726H.4 Free-energy functionals for spin systems and proof of invertibility 727H.5 Extension to quantum spins and density functional theory 729


Appendix I Auxiliary systems and constrained search 731Summary 731

I.1 Auxiliary system to reproduce selected quantities 731I.2 Constrained search with an interacting auxiliary system 732

Exercises 734

Appendix J Derivation of the Luttinger theorem 735Summary 735Select further reading 737Exercises 738

Appendix K Gutzwiller and Hubbard approaches 739Summary 739

K.1 Gutzwiller approach in terms of the wavefunction 740K.2 Hubbard approach in terms of the Green’s function 742K.3 Two scenarios for the Mott transition 747


References 750

Index 806






Preface

Recent progress in the theory and computation of electronic structure is bringing anunprecedented level of capability for research. It is now possible to make quantitativecalculations and provide novel understanding of natural and man-made materials andphenomena vital to physics, chemistry, materials science, as well as many other fields.Electronic structure is indeed an active, growing field with enormous impact, as illustratedby the more than 10,000 papers per year.

Much of our understanding is based on mean-field models of independent electrons,such as Hartree–Fock and other approximations, or density functional theory. The lat-ter is designed to treat ground-state properties of the interacting-electron system, but itis often also used to describe excited states in an independent-electron interpretation.Such approaches can only go so far; many of the most interesting properties of materialsare a result of interaction between electrons that cannot be explained by independent-electron descriptions. Calculations for interacting electrons are much more challengingthan those of independent electrons. However, thanks to developments in theory and meth-ods based on fundamental equations, and thanks to improved computational hardware,many-body methods are increasingly essential tools for a broad range of applications. Withthe present book, we aim to explain the many-body concepts and computational methodsthat are needed for the reader to enter the field, understand the methods, and gain a broadperspective that will enable him or her to participate in new developments.

What sets this book apart from others in the field? Which criteria determine the top-ics included? We want the description to be broad and general, in order to reflect therichness of the field, the generality of the underlying theories, and the wide range of poten-tial applications. The aim is to describe matter all the way from isolated molecules toextended systems. The methods must be capable of computing a wide range of propertiesof diverse materials, and have promise for exciting future applications. Finally, practicalcomputational methods are an important focus for this book.

Choices have to be made since the number of different approaches, their variations,and applications is immense, and the book is meant to be more than an overview. Wetherefore cannot focus on such important areas as quantum chemistry methods, e.g. cou-pled cluster theory and configuration interaction methods, nor do we cover all of thedevelopments in lattice models, or explore the vast field of superconductivity. Rather,we concentrate on three methods: many-body perturbation theory, dynamical mean-field






xviii Preface

theory, and stochastic quantum Monte Carlo methods. Our goals are to present each ofthese widely used methods with its strengths and weaknesses, to bring out their relation-ships, and to illuminate paths for combining these and other methods in order to createmore powerful approaches. We believe that it is a unique feature of this book to have allthree methods addressed in a coherent way: the conceptual structure, the actual methods,and selected applications.

The book is written so that a reader interested in many-body theory and methodsshould not have to go through the details of density functional theory and methods forindependent-particle calculations. Summaries of pertinent material are given in this book,and extensive exposition can be found in the companion book by one of us (R. M. Martin,Electronic Structure: Basic theory and methods, Cambridge University Press, Cambridge,2004, reprinted 2005 and 2008). The two books are complementary and can be readindependently.

The organization of the book follows naturally: Part I (Chapters 1–4) contains back-ground material including history, experimental motivations, and some classic conceptsand models referred to throughout the book. Part II (Chapters 5–8) discusses theoreticalfoundations useful in all many-body treatments: mean-field theories, correlation functionsand Green’s functions, many-body wavefunctions, the concepts of quasi-particles, andfunctionals used in Green’s function methods. Part III contains two distinct but relatedways to use Green’s functions, namely many-body perturbation theory including GW andthe Bethe–Salpeter equation (Chapters 9–15), and dynamical mean-field theory (Chap-ters 16–21). Finally, Part IV (Chapters 22–26) discusses stochastic (quantum Monte Carlo)methods. The descriptions of these three approaches are written so that each can be readindependently. Altogether, we try to clarify concepts, differences, and points of contactbetween the various methods, and possibilities for combining the methods as a basis forfuture progress in understanding and predicting properties of interacting electron systems.






Acknowledgments

This book has benefitted from the support of many colleagues and friends, and from theworking environment provided by institutions and initiatives. The book reflects thoughts,discussions, and struggles for understanding that we have shared with our students andteachers, post docs, and peers. It is not possible to name everyone; people who havecontributed directly are: Alexei Abrikosov, James W. Allen, Ole Krough Andersen, FerdiAryasetiawan, Friedhelm Bechstedt, Arjan Berger, Silke Biermann, Nils Blümmer, FabienBruneval, Michele Casula, Bryan Clark, Rodolfo Del Sole, Thomas P. Devereaux, StefanoDi Sabatino, Jacques Friedel, Christoph Friedrich, Matteo Gatti, Ted Geballe, AntoineGeorges, Rex Godby, Emmanuel Gull, Matteo Guzzo, Walter Harrison, Kristjan Haule,Karsten Held, Markus Holzmann, Mark Jarrell, Borje Johansson, Erik Koch, GabrielKotliar, Wei Ku, Jan Kunes, Giovanna Lani, Alexander I. Lichtenstein, Andrew McMa-han, Andrea Marini, Andrew Millis, Elisa Molinari, Giovanni Onida, Eva Pavarini, WarrenE. Pickett, John Rehr, Claudia Rödl, Pina Romaniello, Mariana Rossi, Alice Ruini, SashiSatpathy, Sergej Savrasov, Richard Scalettar, Fausto Sirotti, Francesco Sottile, Adrian Stan,Zhi-Xun Shen, Mark van Schilfgaarde, Liu Hao Tjeng, Dieter Vollhardt, Jess Wellendorff,Philipp Werner, Jonas Weinen, Weitao Yang, Amy Young, Haijun Zhang, Shiwei Zhang,Jianqiang Sky Zhou.

Of great benefit was the continuous and stimulating scientific exchange in the frameworkof a series of European collaborative projects, culminating in the Nanoquanta network ofexcellence and the European Theoretical Spectroscopy Facility (ETSF).

Our institutions have supported our work through this long project: the Department ofPhysics, the Frederick Seitz Materials Research Laboratory at the University of IllinoisUrbana-Champaign, and the Materials Computation Center, supported by the NationalScience Foundation; the Laboratoire des Solides Irradiés in Palaiseau and its responsibleinstitutions – the Centre National de la Recherche Scientifique (CNRS), the École Poly-technique, and the Commissariat à l’énergie atomique et aux énergies alternatives (CEA);the Department of Applied Physics, Standford University.

Additional support was provided by the Aspen Center of Physics, Agence Nationale pourla Recherche Scientifique (ANR) in France, ICMAB – Universitat Autonoma Barcelona,SIESTA Foundation.

Last not least, we are deeply thankful to our families, friends, and colleagues for theirgreat patience and their continuous encouragements.






Notation

Abbreviations

w.r.t. with respect to+c.c. add complex conjugate of preceding quantityRe, Im real and imaginary partsBZ first Brillouin zone1D one dimension or one-dimensional (similar for 2D and 3D)

Acronyms most used

BSE Bethe–Salpeter equationDFT density functional theoryDMFA various approximations in DMFTDMFT dynamical mean-field theoryEXX exact exchange in DFTGGA generalized gradient approximationGWA GW approximationHF Hartree–FockHFA Hartree–Fock approximationKS Kohn–ShamLDA local density approximationMBPT many-body perturbation theoryQMC quantum Monte CarloRHF restricted Hartree–FockRPA random phase approximationTDDFT time-dependent density functional theoryUHF unrestricted Hartree–Fock

General physical quantities

E energyμ chemical potentialEF Fermi energy (chemical potential at T = 0)T temperature (also computer time)β inverse temperature, 1/kBTZ partition function; same notation for canonical and grand

canonical (also renormalization factor for spectral weight)S or A action






Notation xxi

S entropyF Helmholz free energyG Gibbs free energy (when identified), otherwise G denotes a

Green’s function� grand potential

Time and frequency

t timeτ imaginary time�τ imaginary time stepω frequency (real or complex as specified or as clear from context)z frequency defined in complex planeωn Matsubara frequencieszn position of ωn in complex plane, zn = μ+ iωn

Coordinates and operators

r electron positionσ electron spinx combined space/spin coordinates (r, σ )1 combined space/spin/time coordinates (x1, t1)1 variable to be integratedR N-electron coordinates R = {r1 . . . rN}N number of electronsRI position of ion IZI charge of ion I

O general operator in Schrödinger picture〈O〉 expectation value

OH(t) operator in Heisenberg picture, for static H, OH = eiHtOe−iHt

c† and c creation and annihilation operators for fermionsψ†, ψ and ψ†(t), ψ(t) field operators for fermions; time argument denotes Heisenberg

picture, for static H, ψ†(t) = eiHtψ†e−iHt, ψ(t) = eiHtψe−iHt

b† and b creation and annihilation operators for bosonsn particle number operatorS, m atomic moments or spin, m = 〈S〉

T time-ordering operatorTτ time ordering in imaginary time τA antisymmetrization operator

Tr trace

Tr TrF = − i2π limη→0+Tr

∫∞−∞ dω eiηωF(ω) for T = 0

TrF = Trβ−1 ∑n F(iωn) for T �= 0

for a general function F






xxii Notation

Hamiltonian and eigenstates

H many-body hamiltonianVee electron–electron interactionvc(|ri − rj|) Coulomb interactionEα energy of many-body state labeled αE0 ground-state energy α(R) or �α(R) many-body wavefunction (R denotes ri, i = 1); often subscript

α is omittedV(R) many-body potential energy function

T (R) or (R) trial wavefunctionU(R) N-electron Jastrow factoru(rij) or u(ri, rj) Jastrow factor for a pair of electrons�(R) N-electron phase of the trial wavefunctionEL(R| ) local energy of a trial functionEV variational energy of a trial functionσ 2 variance of the local energy of a trial functionO overlap of trial function with exact wavefunction

H0 independent-particle hamiltonianχm(r) single-particle basis function, m = 1, . . . , Nbasis

hm,m′ or h0m,m′ matrix element of independent-particle hamiltonian

v(r) single-particle potentialvH (r) Hartree potentialψi(r) single-particle wavefunction, i = 1, . . . , Nstates, also orbitals in

a Slater determinantεi independent-particle eigenvalue, i = 1, . . . , Nstates

f (ε) Fermi function

Parameters of model hamiltonians

t, t′ hopping matrix elementsU on-site interaction in Hubbard-type modelsJ intra-atomic exchange interaction, also interatomic Heisenberg

exchange constant

Correlation functions, Green’s functions, response functions, self-energy, etc.

CAB(t) correlation function for quantities A and Bn(r) density (also spin-resolved n(r, σ ))n0(r) ground-state density (also n0(r, σ ))ρ(k) momentum distribution (also ρ(k, σ ))ρ(r, r′) one-electron density matrix (also ρ(r, σ ; r′, σ ′))






Notation xxiii

n(r, r′) pair distribution function (also n(r, σ ; r′, σ ′))g(r, r′) normalized pair distribution (also g(r, σ ; r′, σ ′))S(q, q′), S(k) structure factor (also S(q, σ ; q′, σ ′) or S σσ ′

kk′ )S(ω) dynamic structure factor with similar notation for momenta and

spin

G one-body Green’s function expressed as a function ofarguments, e.g., r, r′, σ , t, τ , ω, z or as a matrix in a basis, Gm,m′

G0 or G0 independent-particle Green’s functionA spectral function� self-energy (irreducible)� vertex functionZ renormalization factor of spectral weight (also partition

function)

G Green’s function for embedded site or cellG0 embedding Green’s function for embedded site or cell� hybridization function for impurity, embedded site or cell

G2 two-particle Green’s functionL two-particle correlation function, L = −G2 + GGχ general linear response function, susceptibilityχ0 general linear response function for independent particlesε dielectric functionW screened interactionP polarizability (irreducible)

Functionals

F[f ] general functional of the function fD domain of a functionalExc[n] exchange–correlation energy in Kohn–Sham theory�[G] interaction functional of Green’s function GF[�] Legendre transform of �[G] [G, W] interaction functional of G and screened interaction WJ (1, 1′) external probe field – non-local in space and time�[J ] grand-potential functional of J�[Q] effective action functional, Legendre transform of �[J ]

Notation for crystals

� volume of cell (primitive cell or supercell)T lattice translations






xxiv Notation

G reciprocal lattice vectorsk or q wavevector in first Brillouin zone

k for electrons; q for interactions, susceptibilitiesψk(r) = eik·ruk(r) Bloch function in crystal, with uk(r) periodicεk eigenvalues that define bands as a function of kGk, �k Green’s function and self-energy as functions of kχq or χ (q) general susceptibility as a function of qGii or G00 on-site Green’s function at site i

Li primitive vector of supercell i = {1, 2, 3}K total wavevector for many-body wavefunctionH(K) K-dependent hamiltonian with periodic eigenvectors

uK(R), R = {r1 . . . rN}

�cf crystal field splittingeg and t2g crystal field states in cubic symmetry

Notation for stochastic methods

S, s, R state vector, i.e., variables describing the instantaneous state ofa random walk

R(n), s(n) the nth many-body configuration in a random walk (n) or τ projected trial wavefunction after n iterations or imaginary

time τ�(s) many-body distribution function to be sampled〈O〉� average of O over distribution �P(s → s′), Ps′s transition probability random walkT(s → s′) trial transition probabilityA(s → s′) acceptance probabilityO estimator for property OO estimated mean value of property OνO variance of property OσO standard error of OκO autocorrelation time for property O in a random walk�τ time step in VMC, DMC, or PIMCP electron permutationG(R, R′), G many-body projectorG(R ← R′) importance-sampled projector G(R) guiding function for importance samplingρ(R, R′;β) many-body density matrix






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