Effective Field Theory for Spontaneously Broken Symmetry

Preface
Contents
Notation and Conventions
	List of Acronyms
	Mathematical Conventions
Part I Prologue
	1 Introduction
		1.1 What Is Effective Field Theory?
		1.2 Broken Symmetry Zoo
		1.3 Structure of This Book
			1.3.1 Further Reading
		References
	2 Our First Model
		2.1 Spontaneous Symmetry Breaking
		2.2 Nambu–Goldstone Boson and Its Interactions
			2.2.1 Linear Parameterization
			2.2.2 Scattering of Nambu–Goldstone Bosons
			2.2.3 Nonlinear Parameterization
		2.3 Low-Energy Effective Field Theory
			2.3.1 Matching
			2.3.2 Eliminating the Heavy Modes
		2.4 Moral Lessons
		References
	3 Generalizations of the Model
		3.1 Relativistic Models with Non-Abelian Symmetry
			3.1.1 Spectrum of Nambu–Goldstone Bosons
			3.1.2 Low-Energy Effective Field Theory
		3.2 Going Nonrelativistic
			3.2.1 Single Schrödinger Field
			3.2.2 Multiple Nambu–Goldstone Fields
		3.3 Moral Lessons
		References
Part II Foundations
	4 Symmetry and Conservation Laws
		4.1 What Is Symmetry?
			4.1.1 Symmetry Transformations
			4.1.2 Object of Symmetry
		4.2 Lagrangian Approach to Symmetry
			4.2.1 Noether's Theorem
			4.2.2 Tensor Conservation Laws
		4.3 Symmetry and Conservation Laws in Hamiltonian Formalism
			4.3.1 Symplectic Formulation of Hamiltonian Dynamics
			4.3.2 Symmetry in Quantum Physics
		References
	5 Spontaneous Symmetry Breaking
		5.1 Physical State and Its Symmetry
			5.1.1 Broken and Unbroken Symmetry
			5.1.2 Symmetrization by Group Averaging
		5.2 Effect of External Perturbations
			5.2.1 Taking the Thermodynamic Limit
			5.2.2 Order Parameter and the Vacuum Manifold
			5.2.3 Intermediate Summary
		5.3 Some Subtle Features of Spontaneous Symmetry Breaking
			5.3.1 Free Schrödinger Field in Finite Volume
			5.3.2 Pathologies of the Infinite-Volume Limit
			5.3.3 Uniqueness of the Finite-Volume Ground State
		References
	6 Nambu–Goldstone Bosons
		6.1 Intuitive Picture
			6.1.1 Redundancy of Order Parameter Fluctuations
			6.1.2 Canonical Conjugation of Nambu–Goldstone Fields
			6.1.3 The Big Picture
		6.2 Goldstone Theorem
			6.2.1 Operator Proof
			6.2.2 Effective Action Proof
		6.3 Classification and Counting
			6.3.1 Independent Order Parameter Fluctuations
			6.3.2 Type-A and Type-B Nambu–Goldstone Bosons
		6.4 Nambu–Goldstone-Like Modes
			6.4.1 Pseudo-Nambu–Goldstone Bosons
			6.4.2 Quasi-Nambu–Goldstone Bosons
			6.4.3 Massive Nambu–Goldstone Bosons
		References
Part III Spontaneously Broken Internal Symmetry
	7 Nonlinear Realization of Symmetry
		7.1 Group Action on Manifolds
		7.2 Classification of Nonlinear Realizations
			7.2.1 Linearization of Group Action
			7.2.2 From Linear Representation to Nonlinear Realization
		7.3 Standard Realization of Symmetry
			7.3.1 Nonlinear Realization on Coset Spaces
			7.3.2 Symmetric Coset Spaces
		7.4 Geometry of the Coset Space
			7.4.1 Canonical and Torsion-Free Connection
			7.4.2 Riemannian Metric
		References
	8 Low-Energy Effective Field Theory
		8.1 Structure of the Effective Lagrangian
			8.1.1 Reminder of the Standard Nonlinear Realization
			8.1.2 Lagrangians with Two Spatial or Two Temporal Derivatives
			8.1.3 Lagrangians with One Temporal Derivative
			8.1.4 Overview of the Lowest-Order Effective Lagrangian
		8.2 Effective Lagrangians from Background Gauge Invariance
			8.2.1 Methodology of Construction of Effective Actions
			8.2.2 Lagrangians Up to Order Two in DerivativeExpansion
			8.2.3 Effects of Explicit Symmetry Breaking
			8.2.4 Coupling to Matter Fields
		8.3 Equation of Motion
			8.3.1 Spectrum of Nambu–Goldstone Bosons Revisited
			8.3.2 More on the Geometry of the Coset Space
		References
	9 Applications to Particle and Condensed-Matter Physics
		9.1 Chiral Perturbation Theory of Mesons
			9.1.1 Power Counting
			9.1.2 Effective Lagrangian
			9.1.3 Interaction with External Fields
			9.1.4 Effects of the Chiral Anomaly
		9.2 Spin Waves in Ferro- and Antiferromagnets
			9.2.1 Power Counting and Effective Lagrangian
				9.2.1.1 Ferromagnets
				9.2.1.2 Antiferromagnets
			9.2.2 Equation of Motion and Magnon Spectrum
			9.2.3 Effects of Symmetry-Breaking Perturbations
			9.2.4 Some Topological Aspects of Ferromagnets
		References
	10 Scattering of Nambu–Goldstone Bosons
		10.1 Adler Zero Revisited
			10.1.1 Generalized Soft Theorem
			10.1.2 Application to Coset Effective Theories
		10.2 Geometric Framework for Scattering Amplitudes
			10.2.1 Geometric Soft Theorem for Nambu–GoldstoneBosons
			10.2.2 Adler Zero or Not?
			10.2.3 Symmetric Coset Spaces
		10.3 Beyond Adler Zero
			10.3.1 Dirac–Born–Infeld Theory
			10.3.2 Galileon and Special Galileon Theory
			10.3.3 Effective Theories with Enhanced Soft Limit from Symmetry
		10.4 Soft Recursion
			10.4.1 Complexified Kinematics
			10.4.2 Recursion Relation for On-Shell Amplitudes
			10.4.3 Soft Bootstrap
		References
Part IV Spontaneously Broken Spacetime Symmetry
	11 Locally Equivalent Symmetries
		11.1 Relations Between Noether Currents
		11.2 Noether Currents from Background Gauging
		11.3 Examples
			11.3.1 Galileon Symmetry
			11.3.2 Spacetime Translations and Rotations
			11.3.3 Galilei Invariance
			11.3.4 Changing the Background: Magnetic Translations
		11.4 Application to Scattering of Nambu–Goldstone Bosons
			11.4.1 Galileon Theory
			11.4.2 Theories with Generalized Shift Symmetry
		References
	12 Nonlinear Realization of Spacetime Symmetry
		12.1 Reminder of Nonlinear Realization of Internal Symmetry
		12.2 Spacetime Symmetry as a Point Transformation
		12.3 Standard Nonlinear Realization
			12.3.1 Summary of the Construction
			12.3.2 Relation to Physics of Broken Spacetime Symmetry
		12.4 Examples
			12.4.1 Lorentz Scalars with Internal Symmetry
			12.4.2 Lorentz Scalars with Scale Invariance
			12.4.3 Lorentz Vector with(out) Lorentz Scalar
			12.4.4 Schrödinger Scalars with Galilei Symmetry
		References
	13 Broken Spacetime Symmetry in Quantum Matter
		13.1 Building Blocks for Construction of Effective Actions
			13.1.1 Maurer–Cartan Form
			13.1.2 Covariant Derivatives of Fields
		13.2 Twisting Order Parameter for Internal Symmetry
			13.2.1 New Features of the Old Setup
			13.2.2 Case Study: Relativistic Superfluids
		13.3 Vector Modes: The Relevant, the Irrelevant and the Unphysical
			13.3.1 The Relevant: Helimagnets
			13.3.2 The Irrelevant: Smectic Liquid Crystals
			13.3.3 The Unphysical: Nonrelativistic Superfluids
			13.3.4 Inverse Higgs Constraints
		13.4 Genuine Breaking of Translation Invariance
			13.4.1 One-Dimensional Modulation of the Order Parameter
			13.4.2 Case Study: Fluctuations of a Domain Wall
			13.4.3 Effective Action from Background Gauge Invariance
			13.4.4 Further Possible Applications
		References
	14 Broken Spacetime Symmetry in Classical Matter
		14.1 Emergent Symmetry of Classical Matter
			14.1.1 Introduction: Spring Model of Elasticity
			14.1.2 Emergent Symmetries of Solids and Fluids
		14.2 Nonlinear Realization of Emergent Symmetry
			14.2.1 Field Variables and Unbroken Symmetry
			14.2.2 Building Blocks for Construction of Effective Actions
		14.3 Effective Field Theory of Classical Matter
			14.3.1 Relativistic Solids
			14.3.2 Nonrelativistic Supersolids
			14.3.3 Nonrelativistic Solids
			14.3.4 Perfect Fluids
		14.4 Coupling Nambu–Goldstone Bosons to Classical Matter
		References
Part V Epilogue
	15 Topics Not Covered in This Book
		15.1 Effects of Nonzero Temperature
		15.2 No-Go Theorems for Spontaneous Symmetry Breaking
		15.3 Topological Aspects of Spontaneous Symmetry Breaking
		15.4 Generalized Symmetries
		References
	16 Some Open Questions
		References
A Elements of Differential Geometry
	A.1 Smooth Manifolds
	A.2 Linear Structures on Manifolds
		A.2.1 Tangent Vectors and Vector Fields
		A.2.2 Tensors and Tensor Fields
	A.3 Maps Between and on Manifolds
		A.3.1 Push-Forward and Pull-Back
		A.3.2 Flow of Vector Fields
		A.3.3 Lie Derivative
	A.4 Exterior Derivative
	A.5 Affine Connection
		A.5.1 Covariant Derivative
		A.5.2 Curvature and Torsion
	A.6 Riemannian Geometry
		A.6.1 Riemannian Metric
		A.6.2 Isometries of Riemannian Metric
		A.6.3 Symmetries of Curvature Tensor
		A.6.4 Geodesic Normal Coordinates
		A.6.5 Hodge Star
	A.7 Integration on Manifolds
		A.7.1 Orientable Manifolds
		A.7.2 Riemannian Manifolds
	A.8 Homology and Cohomology
		A.8.1 Singular Homology
		A.8.2 De Rham Cohomology
	References