Symmetry, Broken Symmetry, and Topology in Modern Physics

Cover
Half-title
Title page
Copyright information
Dedication
Brief Contents
Contents
Preface
Part I Symmetry Groups and Algebras
1 Introduction
2 Some Properties of Groups
2.1 Invariance and Conservation Laws
2.2 Definition of a Group
2.3 Examples of Groups
2.3.1 Additive Group of Integers
2.3.2 Rotation and Translation Groups
2.3.3 Parameterization of Continuous Groups
2.3.4 Permutation Groups
2.4 Subgroups
2.5 Homomorphism and Isomorphism
2.6 Matrix Representations
2.6.1 A Matrix Representation of S[sub(3)]
2.6.2 Dimensionality of Matrix Representations
2.6.3 Linear Operators and Matrix Representations
2.7 Reducible and Irreducible Representations
2.8 Degenerate Multiplet Structure
2.9 Some Examples of Matrix Groups
2.9.1 General Linear Groups
2.9.2 Unitary Groups
2.9.3 Orthogonal Groups
2.9.4 Symplectic Groups
2.10 Group Generators
2.11 Conjugate Classes
2.12 Invariant Subgroups
2.13 Simple and Semisimple Groups
2.14 Cosets and Factor Groups
2.14.1 Left and Right Coset Decompositions
2.14.2 Factor Groups
2.15 Direct Product Groups
2.16 Direct Product of Representations
2.17 Characters of Representations
2.17.1 Character Theorems
2.17.2 Character Tables
Background and Further Reading
Problems
3 Introduction to Lie Groups
3.1 Lie Groups
3.2 Lie Algebras
3.2.1 Invariant Subalgebras
3.2.2 Adjoint Representation of the Algebra
3.3 Angular Momentum and the Group SU(2)
3.3.1 Fundamental Representation of SU(2)
3.3.2 The Cartan–Dynkin Method
3.3.3 Cartan–Dynkin Analysis of SU(2)
3.3.4 The Clebsch–Gordan Series for SU(2)
3.3.5 SU(2) Adjoint Representation
3.4 Isospin
3.4.1 The Neutron–Proton System
3.4.2 Algebraic Structure for Isospin
3.4.3 The U(1) and SU(2) Subgroups of U(2)
3.4.4 Analogy between Angular Momentum and Isospin
3.4.5 The Adjoint Representation of Isospin
3.5 The Importance of Lie Groups in Physics
3.6 Symmetry and Dynamics
3.6.1 Local Gauge Theories
3.6.2 Dynamical Symmetries
Background and Further Reading
Problems
4 Permutation Groups
4.1 Young Diagrams
4.1.1 Two-Particle Young Diagrams
4.1.2 Many-Particle Young Diagrams
4.1.3 A Compact Notation
4.2 Standard Arrangement of Young Tableaux
4.3 Irreducible Representations
4.3.1 Counting Standard Arrangements
4.3.2 The Hook Rule
4.4 Basis Vectors
4.5 Products of Representations
4.5.1 Direct Products
4.5.2 Outer Products
Background and Further Reading
Problems
5 Electrons on Periodic Lattices
5.1 The Direct Lattice
5.1.1 Brevais Lattices
5.1.2 Wigner–Seitz Cells
5.2 The Reciprocal Lattice
5.3 Brillouin Zones
5.4 Bloch’s Theorem
5.5 Electronic Band Structure
5.6 Point Groups
5.6.1 Point Group Operations
5.6.2 The Crystallographic Point Groups
5.7 Example: The Ammonia Molecule
5.7.1 Symmetry Operations
5.7.2 A Matrix Representation
5.7.3 Class Structure
5.7.4 Other Irreducible Representations
5.8 General Lattice Symmetry Classifications
5.9 Space Groups
5.9.1 Elements of the Space Group
5.9.2 Symmorphic Space Groups
Background and Further Reading
Problems
6 The Rotation Group
6.1 Three-Dimensional Rotations
6.2 The SO(2) Group
6.2.1 Generators of SO(2) Rotations
6.2.2 SO(2) Irreducible Representations
6.2.3 Connectedness of the Manifold
6.2.4 Compactness of the Manifold
6.2.5 Invariant Group Integration
6.3 The SO(3) Group
6.3.1 Generators of SO(3)
6.3.2 Matrix Elements of the Rotation Operator
6.3.3 Properties of D-Matrices
6.3.4 Characters for SO(3)
6.3.5 Direct Products of SO(3) Representations
6.3.6 SO(3) Vector-Coupling Coefficients
6.3.7 Properties of SO(3) Clebsch–Gordan Coefficients
6.3.8 3J Symbols
6.3.9 Construction of SO(3) Irreducible Multiplets
6.4 Tensor Operators under Group Transformations
6.5 Tensors for the Rotation Group
6.6 SO(3) Tensor Products
6.7 The Wigner–Eckart Theorem
6.8 The Wigner–Eckart Theorem for SO(3)
6.8.1 Reduced Matrix Elements
6.8.2 Selection Rules
6.9 Relationship of SO(3) and SU(2)
6.9.1 SO(3) and SU(2) Group Manifolds
6.9.2 Universal Covering Group of the SU(2) Algebra
Background and Further Reading
Problems
7 Classification of Lie Algebras
7.1 Adjoint Representations
7.1.1 The Cartan Subalgebra
7.1.2 Raising and Lowering Operators
7.2 The Cartan–Weyl Basis
7.2.1 Semisimple Algebras
7.2.2 Metric Tensor, Semisimplicity, and Compactness
7.3 Structure of the Root Space
7.3.1 Root Space Restrictions
7.3.2 Lengths and Angles for Root Vectors
7.4 Construction of Root Diagrams
7.4.1 Rank-1 and Rank-2 Compact Lie Algebras
7.4.2 An Ordering Prescription for Weights
7.5 Simple Roots
7.6 Dynkin Diagrams
7.6.1 The Cartan Matrix
7.6.2 Constructing All Roots from Dynkin Diagrams
7.6.3 Constructing the Algebra from the Roots
7.7 Dynkin Diagrams and the Simple Algebras
Background and Further Reading
Problems
8 Unitary and Special Unitary Groups
8.1 Generators and Commutators for SU(3)
8.2 SU(3) Casimir Operators
8.3 SU(3) Weight Space
8.3.1 SU(3) Raising and Lowering Operators
8.3.2 SU(3) Irreducible Representations
8.3.3 Dimensionality of SU(3) Irreps
8.3.4 Construction of SU(3) Weight Diagrams
8.4 Complex Conjugate Representations
8.5 Real and Complex Representations
8.6 Unitary Symmetry and Young Diagrams
8.7 Young Diagrams for SU(N)
8.7.1 Two Particles in Two States
8.7.2 Two Particles in Three States
8.7.3 Fundamental and Conjugate Representations
8.8 Dimensionality of SU(N) Representations
8.9 Direct Products of SU(N) Representations
8.10 Weights from Young Diagrams
8.11 Graphical Construction of Direct Products
Background and Further Reading
Problems
9 SU(3) Flavor Symmetry
9.1 Symmetry in Particle Physics
9.1.1 SU(3) Phenomenology and Quarks
9.1.2 Non-Abelian Gauge Symmetries
9.2 Fundamental SU(3) Quark Representations
9.3 SU(3) Flavor Multiplets
9.3.1 Mass Splittings in SU(3) Multiplets
9.3.2 Quark Structure for Mesons and Baryons
9.4 Isospin Subgroups of SU(3)
9.4.1 Subgroup Analysis Using Weight Diagrams
9.4.2 Subgroup Analysis Using Young Diagrams
9.5 Extensions of Flavor SU(3) Symmetry
9.5.1 Higher-Rank Flavor Symmetries
9.5.2 SU(6) Flavor–Spin Symmetry
9.5.3 Baryons and Mesons under SU(6) Symmetry
Background and Further Reading
Problems
10 Harmonic Oscillators and SU(3)
10.1 The 3D Quantum Oscillator
10.1.1 Eigenvalues
10.1.2 Wavefunctions
10.1.3 Unitary Symmetry
10.1.4 Angular Momentum Subgroup
10.1.5 SO(3) Transformation Properties
10.1.6 Group Structure
10.1.7 Many-Body Operators
10.2 SU(3) and the Nuclear Shell Model
10.3 SU(3) Classification of SD Shell States
10.3.1 Classification Strategy
10.3.2 Orbital and Spin–Isospin Symmetry
10.3.3 Permutation Symmetry
10.3.4 Example: Two Particles in the SD Shell
10.4 SU(2) Subgroups and Intrinsic States
10.4.1 Weight Space Operators and Diagrams
10.4.2 Angular Momentum Content of Multiplets
10.5 Collective Motion in the Nuclear SD Shell
10.5.1 Hamiltonian
10.5.2 Group-Theoretical Solution
10.5.3 The Theoretical Spectrum
Background and Further Reading
Problems
11 SU(3) Matrix Elements
11.1 Clebsch–Gordan Coefficients for SU(3)
11.2 Constructing SU(3) Clebsch–Gordan Coefficients
11.3 Matrix Elements of Generators
11.4 Isoscalar Factors
11.4.1 Racah Factorization Lemma
11.4.2 Evaluating and Using Isoscalar Factors
11.5 SU(3)⊃SO(3) Tensor Operators
11.6 The SU(3) Wigner–Eckart Theorem
11.7 Structure of SU(3) Matrix Elements
11.8 The Gell-Mann, Okubo Mass Formula
11.9 SU(3) Oscillator Reduced Matrix Elements
11.9.1 Spherical Operators
11.9.2 Matrix Elements for Creation and Annihilation Operators
11.9.3 Electromagnetic Transitions in the SD Shell
11.10 Lie Algebras and Many-Body Systems
Background and Further Reading
Problems
12 Introduction to Non-Compact Groups
12.1 Review of the Compact Group SU(n)
12.2 The Non-Compact Group SU(l,m)
12.2.1 Signature of the Metric
12.2.2 Parameter Space for SU(1,1)
12.3 The Non-Compact Group SO(l,m)
12.4 Euclidean Groups
12.4.1 The Euclidean Group E[sub(3)] for 3D Space
12.4.2 The Euclidean Group E[sub(2)] for 2D Space
12.4.3 Semidirect Product Groups
12.4.4 Algebraic Properties of E[sub(2)]
12.4.5 Invariant Subgroup of Translations
12.5 Method of Induced Representations for E[sub(2)]
12.5.1 Generating the Representation
12.5.2 Significance of the Abelian Invariant Subgroup
Background and Further Reading
Problems
13 The Lorentz Group
13.1 Spacetime Tensors
13.1.1 A Covariant Notation
13.1.2 Tensor Transformation Laws
13.2 Lorentz Transformations
13.2.1 Lorentz Boosts as Minkowski Rotations
13.2.2 Generators of Boosts and Rotations
13.2.3 Commutation Algebra for the Lorentz Group
13.3 Classification of Lorentz Transformations
13.3.1 The Four Pieces of the Full Lorentz Group
13.3.2 Improper Lorentz Transformations
13.3.3 Lightcone Classification of Minkowski Vectors
13.4 Properties of the Lorentz Group
13.5 The Lorentz Group and SL(2,C)
13.5.1 A Mapping between 4-Vectors and Matrices
13.5.2 The Universal Covering Group of SO(3,1)
13.6 Spinors and Lorentz Transformations
13.6.1 SU(2)×SU(2) Representations of the Lorentz Group
13.6.2 Two Inequivalent Spinor Representations
13.7 Space Inversion for the Lorentz Group
13.7.1 Action of Parity on Generators and Representations
13.7.2 General and Self-Conjugate Representations
13.8 Parity and 4-Spinors
13.9 Higher-Dimensional Lorentz Representations
13.10 Non-Unitarity of Representations
13.11 Meaning of Non-Unitary Representations
Background and Further Reading
Problems
14 Lorentz-Covariant Fields
14.1 Lorentz Covariance of Maxwell’s Equations
14.1.1 Scalar and Vector Potentials
14.1.2 Gauge Transformations
14.1.3 Manifestly Covariant Form of the Maxwell Equations
14.2 The Dirac Equation
14.2.1 Lorentz-Boosted Spinors
14.2.2 A Lorentz-Covariant Notation
14.3 Dirac Bilinear Covariants
14.3.1 Covariance of the Dirac Equation
14.3.2 Transformation Properties of Bilinear Products
14.4 Weyl Equations and Massless Fermions
14.5 Chiral Invariance
14.5.1 Helicity States for Fermions
14.5.2 Dirac Equation in Pauli–Dirac Representation
14.5.3 Helicity and Chirality for Dirac Fermions
14.5.4 Projection Operators for Chiral Fermions
14.5.5 Interactions and Chiral Symmetry
14.6 The Majorana Equation
14.6.1 Dirac and Majorana Masses
14.6.2 Neutrinoless Double β-Decay
14.7 Summary: Possible Spinor Types
14.8 Spinor Symmetry in the Weak Interactions
14.8.1 The Left Hand of the Neutrino
14.8.2 Violation of Parity P
14.8.3 C, CP, and T Symmetries
14.8.4 A More Complete Picture
Background and Further Reading
Problems
15 Poincaré Invariance
15.1 The Poincaré Multiplication Rule
15.2 Generators of Poincaré Transformations
15.2.1 Proper Lorentz Transformations
15.2.2 Four-Dimensional Spacetime Translations
15.2.3 Commutators for Poincaré Generators
15.3 Representation Theory of the Poincaré Group
15.3.1 Casimir Operators for the Poincaré Group
15.3.2 Classification of Poincaré States
15.3.3 Method of Induced Representations
15.4 Massive Representations of the Poincaré Group
15.4.1 Quantum Numbers for Massive States
15.4.2 Action of the Poincaré Group on Massive States
15.4.3 Summary: Representations for Massive States
15.5 Massless Representations
15.5.1 The Standard Lightlike Vector
15.5.2 Lie Algebra of the Little Group
15.5.3 Quantum Numbers for Massless States
15.6 Mass and Spin for Poincaré Representations
15.7 Lorentz and Poincaré Representations
15.7.1 Operators for Relativistic Quantum Fields
15.7.2 Wave Equations for Quantum Fields
15.7.3 Plane-Wave Expansion of the Fields
15.7.4 The Relationship of Fields and Particles
15.7.5 Symmetry and the Wave Equation
Background and Further Reading
Problems
16 Gauge Invariance
16.1 Relativistic Quantum Field Theory
16.1.1 Quantization of Classical Fields
16.1.2 Symmetries of the Classical Action
16.1.3 Lagrangian Densities for Free Fields
16.1.4 Euler–Lagrange Field Equations
16.2 Conserved Currents and Charges
16.2.1 Noether’s Theorem
16.2.2 Conserved Charges
16.2.3 Symmetries for Interacting Fields
16.2.4 Partially Conserved Currents
16.3 Gauge Invariance in Quantum Mechanics
16.4 Gauge Invariance and the Photon Mass
16.5 Quantum Electrodynamics
16.5.1 Global U(1) Gauge Invariance
16.5.2 Local U(1) Gauge Invariance
16.5.3 Gauging the U(1) Symmetry
16.6 Yang–Mills Fields
16.6.1 Non-Abelian Gauge Invariance
16.6.2 Covariant Derivatives
16.6.3 Non-Abelian Generalization of QED
16.6.4 Properties of Non-Abelian Gauge Fields
Background and Further Reading
Problems
Part II Broken Symmetry
17 Spontaneous Symmetry Breaking
17.1 Modes of Symmetry Breaking
17.2 Explicit Symmetry Breaking
17.3 The Vacuum and Hidden Symmetry
17.4 Spontaneously Broken Discrete Symmetry
17.4.1 Symmetry in the Wigner Mode
17.4.2 Spontaneously Broken Symmetry
17.4.3 Summary of Spontaneously Broken Discrete Symmetry
17.5 Spontaneously Broken Continuous Symmetry
17.5.1 Symmetric Classical Vacuum
17.5.2 Hidden Continuous Symmetry
17.5.3 The Goldstone Theorem
17.5.4 The Stability Subgroup
Background and Further Reading
Problems
18 The Higgs Mechanism
18.1 Photons and the Higgs Loophole
18.2 The Abelian Higgs Model
18.2.1 Lagrangian Density
18.2.2 Symmetry Breaking
18.2.3 Understanding the Higgs Mechanism
18.3 Vacuum Screening Currents
18.3.1 Gauge Invariance and Mass
18.3.2 Screening Currents and Effective Mass
18.3.3 Atomic Screening Currents
18.3.4 The Meissner Effect and Massive Photons
18.3.5 Gauge Invariance and Longitudinal Polarization
18.4 The Higgs Boson
Background and Further Reading
Problems
19 The Standard Model
19.1 The Standard Electroweak Model
19.1.1 Guidance from Data
19.1.2 The Gauge Group
19.1.3 Electroweak Lagrangian Density
19.1.4 The Electroweak Higgs Mechanism
19.1.5 Particle Spectrum
19.2 Quantum Chromodynamics
19.2.1 A Color Gauge Theory
19.2.2 The QCD Lagrangian Density
19.2.3 Symmetries of the QCD Lagrangian Density
19.2.4 Asymptotic Freedom and Confinement
19.2.5 Exotic Hadrons and Glueballs
19.3 The Gauge Theory of Fundamental Interactions
Background and Further Reading
Problems
20 Dynamical Symmetry
20.1 The Microscopic Dynamical Symmetry Method
20.1.1 Solution Algorithm
20.1.2 Validity and Utility of the Approach
20.1.3 Spontaneously Broken Symmetry and Dynamical Symmetry
20.1.4 Kinematics and Dynamics
20.2 Monolayer Graphene in a Strong Magnetic Field
20.2.1 Electronic Dispersion in Monolayer Graphene
20.2.2 Landau Levels for Massless Dirac Electrons
20.2.3 SU(4) Quantum Hall Ferromagnetism
20.2.4 Fermion Dynamical Symmetries for Graphene
20.2.5 Graphene SO(8) Dynamical Symmetries
20.2.6 Generalized Coherent States for Graphene
20.2.7 Physical Interpretation of the Energy Surfaces
20.2.8 Quantum Phase Transitions in Graphene
20.3 Universality of Emergent States
20.3.1 Topological and Algebraic Constraints
20.3.2 Analogy with General Relativity
20.3.3 Analogy with Renormalization Group Flow
Background and Further Reading
Problems
21 Generalized Coherent States
21.1 Glauber Coherent States
21.2 Symmetry and Coherent Electromagnetic States
21.2.1 Quantum Optics Hamiltonian
21.2.2 Symmetry of the Hamiltonian
21.2.3 Hilbert Space
21.2.4 Stability Subgroup
21.2.5 Coset Space
21.2.6 The Coherent State
21.3 Construction of Generalized Coherent States
21.4 Atoms Interacting with Classical Radiation
21.5 Fermion Coherent States
Background and Further Reading
Problems
22 Restoring Symmetry by Projection
22.1 Rotational Symmetry in Atomic Nuclei
22.2 The Method of Generator Coordinates
22.2.1 Generator Coordinates and Generating Functions
22.2.2 The Hill–Wheeler Equation
22.3 Angular Momentum Projection
22.3.1 The Rotation Operator and its Representations
22.3.2 The Angular Momentum Projection Operator
22.3.3 Solving the Eigenvalue Equation
22.4 Particle Number Projection
22.4.1 Violation of Particle Number in BCS Theory
22.4.2 Bogoliubov Quasiparticles
22.4.3 The Particle Number Projection Operator
22.5 Parity Projection
22.5.1 The Parity Transformation
22.5.2 Breaking Parity Spontaneously
22.5.3 The Parity Projection Operator
22.6 Spin and Momentum Projection for Electrons
22.6.1 Hartree–Fock Approximation for the Hubbard Model
22.6.2 Spin and Momentum Projection in the Hubbard Model
Background and Further Reading
Problems
23 Quantum Phase Transitions
23.1 Classical and Quantum Phases
23.1.1 Thermal and Quantum Fluctuations
23.1.2 Quantum Critical Behavior
23.2 Classification of Phase Transitions
23.3 Classical Second-Order Phase Transitions
23.3.1 Critical Exponents
23.3.2 Universality
23.4 Continuous Quantum Phase Transitions
23.4.1 Order Only at Zero Temperature
23.4.2 Order Also at Finite Temperature
23.5 Quantum to Classical Crossover
23.5.1 The Classical–Quantum Mapping
23.5.2 Optimal Dimensionality
23.5.3 Quantum versus Classical Phase Transitions
23.6 Example: Ising Spins in a Transverse Field
23.6.1 Hamiltonian
23.6.2 Ground States and Quasiparticle States for g → 0
23.6.3 Ground States and Quasiparticle States for g → ∞
23.6.4 Competing Ground States
23.6.5 The Quantum Critical Region
23.6.6 Phase Diagram
23.7 Dynamical Symmetry and Quantum Phases
23.7.1 Quantum Phases in Superconductors
23.7.2 Unique Perspective of Dynamical Symmetries
23.7.3 Quantum Phases and Insights from Symmetry
Background and Further Reading
Problems
Part III Topology and Geometry
24 Topology, Manifolds, and Metrics
24.1 Basic Concepts of Topology
24.1.1 Discrete Categories Distinguished Qualitatively
24.1.2 The Nature of Topological Proofs
24.1.3 Neighborhoods
24.2 Topology and Topological Spaces
24.2.1 Formal Definition of a Topology
24.2.2 Continuity
24.2.3 Compactness
24.2.4 Connectedness
24.2.5 Homeomorphism
24.3 Topological Invariants
24.3.1 Compactness Is a Topological Invariant
24.3.2 Connectedness Is a Topological Invariant
24.3.3 Dimensionality Is a Topological Invariant
24.4 Homotopies
24.4.1 Homotopic Equivalence Classes
24.4.2 Homotopy Classes Are Topological Invariants
24.4.3 The First Homotopy Group
24.4.4 Higher Homotopy Groups
24.5 Manifolds and Metric Spaces
24.5.1 Differentiable Manifolds
24.5.2 Metric Spaces
Background and Further Reading
Problems
25 Topological Solitons
25.1 Models in (1+1) Dimensions
25.1.1 Equations of Motion
25.1.2 Vacuum States and Boundary Conditions
25.1.3 Topological Charges
25.1.4 Soliton Solutions in (1+1) Dimensions
25.2 Solitons in (2+1) and (3+1) Dimensions
25.2.1 Homotopy Groups
25.2.2 Mapping Spheres to Spheres
25.3 Yang–Mills Fields and Instantons
25.3.1 Solitons in the Euclidean Yang–Mills Field
25.3.2 Boundary Conditions
25.3.3 Topological Classification of Solutions
25.3.4 Physical Interpretation of Instantons
Background and Further Reading
Problems
26 Geometry and Gauge Theories
26.1 Parallel Transport
26.1.1 Flat and Curved Manifolds
26.1.2 Connections and Covariant Derivatives
26.1.3 Curvature and Parallel Transport
26.2 Absolute Derivatives
26.3 Parallel Transport in Charge Space
26.4 Fiber Bundles and Gauge Manifolds
26.4.1 Tangent Spaces and Tangent Bundles
26.4.2 Fiber Bundles
26.5 Gauge Symmetry on a Spacetime Lattice
26.5.1 Path-Dependent Gauge Representations
26.5.2 Lattice Gauge Symmetries
Background and Further Reading
Problems
27 Geometrical Phases
27.1 The Aharonov–Bohm Effect
27.1.1 Experimental Setup
27.1.2 Analysis of Magnetic Fields
27.1.3 Phase of the Electron Wavefunction
27.1.4 Topological Origin of the Aharonov–Bohm Effect
27.2 The Berry Phase
27.2.1 Fast and Slow Degrees of Freedom
27.2.2 The Berry Connection
27.2.3 Trading the Connection for a Phase
27.2.4 Berry Phases
27.2.5 Berry Curvature
27.3 An Electron in a Magnetic Field
27.4 Topological Implications of Berry Phases
Background and Further Reading
Problems
28 Topology of the Quantum Hall Effect
28.1 The Classical Hall Effect
28.1.1 Hall Effect Measurements
28.1.2 Quantization of the Hall Effect
28.2 Landau Levels for Non-Relativistic Electrons
28.2.1 Hamiltonian and Schrödinger Equation
28.2.2 Landau Levels and Density of States
28.3 The Integer Quantum Hall Effect
28.3.1 Understanding the Integer Quantum Hall Effect
28.3.2 Disorder and the Integer Quantum Hall State
28.3.3 Edge States and Conduction
28.4 Topology and Integer Quantum Hall Effects
28.4.1 Berry Phases and Adiabatic Curvature
28.4.2 Chern Numbers
28.5 The Fractional Quantum Hall Effect
28.5.1 Properties of the Fractional Quantum Hall State
28.5.2 Fractionally Charged Quasiparticles
28.5.3 Nature of the Edge States
28.5.4 Topology and Fractional Quantum Hall States
Background and Further Reading
Problems
29 Topological Matter
29.1 Topology and the Many-Body Paradigm
29.1.1 Adiabatic Continuity
29.1.2 Spontaneous Symmetry Breaking
29.1.3 Beyond the Landau Picture
29.2 Berry Phases and Brillouin Zones
29.3 Topological States and Symmetry
29.4 Topological Insulators
29.4.1 The Quantum Spin Hall Effect
29.4.2 The Z[sub(2)] Topological Index
29.5 Weyl Semimetals
29.5.1 A Topological Conservation Law
29.5.2 Realization of a Weyl Semimetal
29.6 Majorana Modes
29.6.1 The Dirac Equation in Condensed Matter
29.6.2 Quasiparticles and Anti-Quasiparticles
29.7 Topological Superconductors
29.7.1 Topological Majorana Fermions
29.7.2 Fractionalization of Electrons
29.8 Fractional Statistics
29.8.1 Anyon Statistics
29.8.2 The Braid Group
29.8.3 Abelian and Non-Abelian Anyons
29.9 Quantum Computers and Topological Matter
29.9.1 Qubits and Quantum Information
29.9.2 The Problem of Decoherence
29.9.3 Topological Quantum Computation
Background and Further Reading
Problems
Part IV A Variety of Physical Applications
30 Angular Momentum Recoupling
30.1 Recoupling of Three Angular Momenta
30.1.1 6J Coefficients
30.1.2 Racah Coefficients
30.2 Matrix Elements of Tensor Products
30.3 Recoupling of Four Angular Momenta
30.3.1 9J Coefficients
30.3.2 Transformation Between L–S and J–J Coupling
30.3.3 Matrix Element of an Independent Tensor Product
30.3.4 Matrix Element of a Scalar Product
Background and Further Reading
Problems
31 Nuclear Fermion Dynamical Symmetry
31.1 The Ginocchio Model
31.2 The Fermion Dynamical Symmetry Model
31.2.1 Dynamical Symmetry Generators
31.2.2 The FDSM Dynamical Symmetries
31.2.3 FDSM Irreducible Representations
31.2.4 Quantitative FDSM Calculations
31.3 The Interacting Boson Model
Background and Further Reading
Problems
32 Superconductivity and Superfluidity
32.1 Conventional Superconductors
32.2 Unconventional Superconductors
32.3 The SU(4) Model of Non-Abelian Superconductors
32.3.1 The SU(4) Algebra
32.3.2 The SU(4) Collective Subspace
32.3.3 The Dynamical Symmetry Hamiltonian
32.3.4 The SU(4) Dynamical Symmetry Limits
32.3.5 The SO(4) Dynamical Symmetry Limit
32.3.6 The SU(2) Dynamical Symmetry Limit
32.3.7 The SO(5) Dynamical Symmetry Limit
32.3.8 Conventional and Unconventional Superconductors
32.4 Some Implications of SU(4) Symmetry
32.4.1 No Double Occupancy
32.4.2 Quantitative Gap and Phase Diagrams
32.4.3 Coherent State Energy Surfaces
32.4.4 Fundamental SU(4) Instabilities
32.4.5 Origin of High Critical Temperatures
32.4.6 Universality of Dynamical Symmetry States
Background and Further Reading
Problems
33 Current Algebra
33.1 The CVC and PCAC Hypotheses
33.1.1 Current Algebra and Chiral Symmetry
33.1.2 The Partially Conserved Axial Current
33.2 The Linear σ-Model
33.2.1 The Particle Spectrum
33.2.2 Explicit Breaking of Chiral Symmetry
Background and Further Reading
Problems
34 Grand Unified Theories
34.1 Evolution of Fundamental Coupling Constants
34.2 Minimal Criteria for a Grand Unified Group
34.3 The SU(5) Grand Unified Theory
34.4 Beyond Simple GUTs
Background and Further Reading
Problems
Appendix A Second Quantization
A.1 Symmetrized Many-Particle Wavefunctions
A.1.1 Bosonic and Fermionic Wavefunctions
A.1.2 Slater Determinants
A.2 Dirac Notation
A.2.1 Bras, Kets, and Bra-Ket Pairs
A.2.2 Bras and Kets as Row and Column Vectors
A.2.3 Linear Operators Acting on Bras and Kets
A.3 Occupation Number Representation
A.3.1 Creation and Annihilation Operators
A.3.2 Basis Transformations
A.3.3 Many-Particle Vector States
A.3.4 One-Body and Two-Body Operators
Appendix B Natural Units
B.1 The Advantage of Natural Units
B.2 Natural Units in Quantum Field Theory
Appendix C Angular Momentum Tables
Appendix D Lie Algebras
References
Index