Advanced Quantum Condensed Matter Physics: One-Body, Many-Body, and Topological Perspectives

840 Pages·2020·33.554 MB·English

by  Michael El-Batanouny

ADVANCED QUANTUM CONDENSED MATTER PHYSICS
Condensed matter physics has fast become the largest discipline within physics. Based
on an established course, this comprehensive textbook covers one-body, many-body, and
topological perspectives. It is the first textbook that presents a comprehensive coverage
of topological aspects of condensed matter as a distinct, yet integrated, component. It
covers topological fundamentals and their connection to physics, introduces Berry phase
and Chern numbers, describes general topological features of band structures, and
delineatesitsclassification.ApplicationsasmanifestinthequantumHalleffect,topological
insulators, and Weyl semimetal are presented. Modern topics of current interest are
explored in-depth, helping students prepare for cutting-edge research. These include
one-electron band theory, path integrals, and coherent states functional integrals as well
as Green and Matsubara functions, spontaneous symmetry breaking, superfluidity, and
superconductivity.Multiplechapterscoveringquantummagnetismarealsoincluded.With
end-of-chapter exercises throughout, it is ideal for graduate students studying advanced
condensedmatterphysics.
michael el-batanouny is a professor of physics at Boston University. He is the author
of Symmetry and Condensed Matter Physics: A Computational Approach (Cambridge
University Press, 2008). His research area is in surface physics, where he has an
international reputation in the topics of nonlinear surface physics, surface dynamics,
surface magnetism, and more recently topological aspects of surfaces, especially in
electronphonon-relatedphenomena.

ADVANCED QUANTUM CONDENSED
MATTER PHYSICS
One-Body, Many-Body, and Topological Perspectives
MICHAEL EL-BATANOUNY
BostonUniversity
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DOI:10.1017/9781108691291
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Contents
Preface pagexiii
PartOne One-ElectronTheory 1
1 Preliminaries 3
1.1 PeriodicLattices,BrillouinZones,andBloch’sTheorem 3
1.2 EnergeticsandDensityScalingforanElectronGas 8
1.3 NoninteractingFreeElectronGas(SommerfeldGas) 10
1.4 DispersionRelationsandtheDensityofStates,D(E) 11
1.5 Born–OppenheimerApproximation 13
1.6 Time-ReversalSymmetry 16
1.7 Appendix:TheAdiabaticApproximation 27
Exercises 30
2 ElectronsandBandTheory:FormalismintheOne-ElectronApproximation 33
2.1 TheMany-BodyProblemandOne-ElectronApproximation 33
2.2 TheHartree–FockModel 35
2.3 TheDensityFunctionalFormalism 44
Exercises 58
3 ElectronsandBandTheory:MethodsofEnergy-BandCalculations 63
3.1 TheCrystalPotential 63
3.2 MethodsofElectronBandCalculations 67
3.3 Pseudopotentials 92
Exercises 106
4 ElectronsandBandTheory:EffectsofSpin–OrbitInteractions 115
4.1 Introduction 115
4.2 SOCinSolid-StatePhysics 118
4.3 Thek·pMethodandtheEffectiveMassTensor 125
4.4 RashbaSpin–OrbitInteractionsinTwo-DimensionalElectronSystems 134
Exercises 142
vii
viii Contents
5 LinearResponseandtheDielectricFunction 146
5.1 Introduction 146
5.2 LinearResponse 146
5.3 TheDielectricFunction:LinearResponseto
ElectromagneticPerturbations 155
Exercises 164
6 PhononsandLatticeDynamics 166
6.1 Introduction 166
6.2 CouplingofPhononstoElectrons 167
6.3 IonicandElectronicContributionstoPhononEnergies 168
6.4 ElectronicContributiontoPhononEnergies 171
6.5 TheDynamicalMatrix 174
6.6 ElectronicEffectsonPhononsinNormalMetals 178
6.7 ElectronicEffectsonPhononsinInsulatorsandSemiconductors 179
6.8 Measurement of the Structure and Dynamics of Crystals:
ParticleScatteringbyCrystallineSolids 185
Exercises 195
7 Dimensionality,Susceptibility,andInstabilities 200
7.1 Dimensionality,Susceptibility,andNesting 200
7.2 PeierlsInstabilityandPeierlsTransition 206
7.3 Electron–PhononCouplingandtheKohnAnomaly 209
Exercises 215
PartTwo TopologicalPhases 217
8 TopologicalAspectsofCondensedMatterPhysics:AHistoricalPerspective 219
9 TopologicalPreliminaries 224
9.1 DefiningImportantBuildingBlocks 224
9.2 TangentandCotangentSpaces 227
9.3 FiberBundles 232
9.4 CovariantDerivativesandCommutatorsofVectorFields 239
9.5 ConnectionsandParallelTransport 240
9.6 RelevancetothePhysicsofTopologicalPhases 250
Exercises 252
10 Berry-ology 254
10.1 Introduction 254
10.2 AasaBerryConnection 256
10.3 PedagogicalExample1:ATwo-LevelSystem 265
10.4 PedagogicalExample2:MolecularAharonov–BohmEffect 269
Exercises 282