Quantum Field Theory: Feynman Path Integrals and Diagrammatic Techniques in Condensed Matter

Cover
Half Title
Title Page
Copyright Page
Contents
Preface
About the Author
1. Symmetry Requirements in QFT
	1.1.Second Quantization
		1.1.1.Fock Space
		1.1.2.Creation and Annihilation Operators
		1.1.3.(Anti)Commutation Relations
		1.1.4.Change of Basis in Second Quantization
		1.1.5.Quantum Field Operators
		1.1.6.Operators in Second-Quantized Form
			1.1.6.1.One-Body Operator
			1.1.6.2.Two-Body Operator
2. Coherent States
	2.1.Coherent States for Bosons
	2.2.Coherent States and Overcompleteness
		2.2.1.Overcompleteness of Coherent States
		2.2.2.Overlap of Two Coherent States
		2.2.3.Overcompleteness Condition
		2.2.4.Closure Relation via Schur’s Lemma
		2.2.5.Normal-Ordered Operators
		2.2.6.The Trace of an Operator
	2.3.Grassmann Algebra and Fermions
		2.3.1.Grassmann Algebra
			2.3.1.1.Differentiation over Grassmann Variables
			2.3.1.2.Exponential Function of Grassmann Numbers
			2.3.1.3.Involution of Grassmann Numbers
			2.3.1.4.Bilinear Form of Operators
			2.3.1.5.Berezin Integration
			2.3.1.6.Grassmann Delta Function
			2.3.1.7.Scalar Product of Grassmann Algebra
		2.3.2.Fermions
	2.4.Fermions and Coherent States
		2.4.1.Coherent State Overcompleteness Relation Proof
		2.4.2.Trace of a Physical Quantity
		2.4.3.Functional Integral Time-Ordered Property
	2.5.Gaussian Integrals
		2.5.1.Multidimensional Gaussian Integral
		2.5.2.Multidimensional Complex Gaussian Integral
		2.5.3.Multidimensional Grassmann Gaussian Integral
	2.6.Wick Theorem for Multidimensional Grassmann Integrals
		2.6.1.Wick Theorem
3.Fermionic and Bosonic Path Integrals
	3.1.Coherent State Path Integrals
	3.2.Noninteracting Particles
		3.2.1.Bare Partition Function
		3.2.2. Inverse Matrix of S(α)
	3.3.Bare Green’s Function via Generating Functional
		3.3.1.Generating Functional
	3.4.Single-Particle Green’s Function
		3.4.1.Matsubara Green’s Function
	3.5.Noninteracting Green’s Function
	3.6 Average Value of a Functional
4.Perturbation Theory and Feynman Diagrams
	4.1.Representation as Diagrams
	4.2.Generating Functionals
	4.3.Wick Theorem
	4.4.Perturbation Theory
		4.4.1.Linked Cluster Theorem
		4.4.2.Green’s Function Generating Functional
		4.4.3.Green’s Functions
			4.4.3.1.Zeroth Order
			4.4.3.2.First Order
			4.4.3.3.Second Order
5. (Anti)Symmetrized Vertices
	5.1.Fully (Anti)Symmetrized Vertices
6.Generating Functionals
	6.1.Connected Green’s Functions
	6.2.General Case
	6.3.Dyson-Schwinger Equations
	6.4.Effective Action For 1PI Green’s Functions
		6.4.1.Normal Systems
		6.4.2.Self-Energy and Dyson Equation
			6.4.2.1.Self-Energy and Dyson Equation
		6.4.3.Higher-Order Vertices
		6.4.4.General Case
		6.4.5.Luttinger-Ward Functional and 2PI Vertices
			6.4.5.1.Normal Systems
			6.4.5.2.The Self-Consistent Dyson Equation
			6.4.5.3.Diagrammatic Interpretation of LWF
			6.4.5.4.2PI Vertices and Bethe-Salpeter Equation
			6.4.5.5.Bethe-Salpeter Equation
7.Random Phase Approximation (RPA)
	7.1.Path Integral Formalism
		7.1.1.Quantum Three-Dimensional Coulomb Gas
		7.1.2.Translationally Invariant System
	7.2 RPA Functional Integral
		7.2.1.Gaussian Fluctuations
			7.2.1.1.Integration over Grassmann Variables
			7.2.1.2.Fermionic Determinant Gaussian Expansion
			7.2.1.3.Diagrammatic Interpretation of the RPA
			7.2.1.4.Saddle-Point Approximation
			7.2.1.5.Lindhard Function and Plasmon Oscillations
			7.2.1.6.Particle-Hole Pair Excitation
			7.2.1.7.Lindhard Formula
			7.2.1.8.Spectral Function
			7.2.1.9.Plasma Oscillations And Landau Damping
			7.2.1.10.Thomas-Fermi Screening
			7.2.1.11.Friedel Oscillations
			7.2.1.12.Dynamic Polarization Function
			7.2.1.13.Ground-State Energy in the RPA
			7.2.1.14.Compressibility
			7.2.1.15.One-Particle Property: Hartree-Fock Theory
8.Phase Transitions and Critical Phenomena
	8.1.Landau Theory of Phase Transition
	8.2.Entropy and Specific Heat
	8.3.External Field Effect on a Phase Transition
	8.4.Ginzburg-Landau Theory
	8.5.The Scaling Hypothesis
	8.6.Identities from the d-Dimensional Space
	8.7.Energy Fluctuation
9.Weakly Interacting Bose Gas
	9.1.Bose-Einstein Condensation
	9.2.Bogoliubov Transformation
	9.3.Nonideal Bose Gas Path Integral Formalism
		9.3.1.Beliaev-Dyson Equations
10.Superconductivity Theory
	10.1.BCS Superconductivity Theory
		10.1.1.Electron-Phonon Interaction in a Solid State
		10.1.2.Effective Four-Fermion BCS Theory
		10.1.3.Effective Action Functional
		10.1.4.Critical Temperature
	10.2.Mean-Field Theory
	10.3.Green’s Function via Bogoliubov Coefficients
	10.4.The BCS Ground State
	10.5.Gauge Invariance
	10.6.Diagrammatic Approach to Superconductivity
		10.6.1.Ladder Approximation
		10.6.2.Bethe-Salpeter Equation
		10.6.3.Cooper Instability
			10.6.3.1.Finite Temperature Calculation
		10.6.4.Small Momentum Transfer Vertex Function
		10.6.5.Ward Identities: Gauge Invariance
		10.6.6.Galilean Invariance
		10.6.7.Response on Vector Potential
11.Path Integral Approach to the BCS Theory
	11.1.Two-Component Fermi Gas Action Functional
	11.2.Hubbard-Stratonovich Fields
		11.2.1.Nambu-Gorkov Representation
		11.2.2.Pairing-Order Parameter Effective Action
		11.2.3.Reciprocal Space
	11.3.Saddle-Point Approximation
	11.4.Generalized Correlation Functions
	11.5.Condensate Fraction
	11.6.Pair Correlation Length
	11.7.Improvement of the Saddle-Point Solution
	11.8.Fluctuation Partition Function
	11.9.Fluctuation Bosonic Partition Function
	11.10.Number Equation Fluctuation Contributions
	11.11.Collective Mode Excitations
12.Green’s Function Averages over Impurities
	12.1.Scattering Potential and Disordered System
	12.2.Disorder Diagrams
	12.3.Perturbation Series T-Matrix Expansion
	12.4.T-Matrix Expansion
	12.5.Disorder Averaging
	12.6.Green’s Function Perturbation Series
	12.7.Quenched Average and White Noise Potential
	12.8.Average over Impurities’ Locations
	12.9.Disorder Average Green’s Function
	12.10.Disorder Diagrams
	12.11.Gorkov Equation with Impurities
		12.11.1.Properties of Homogeneous Superconductors
13.Classical and Quantum Theory of Magnetism
	13.1 Classical Theory of Magnetism
		13.1.1.Molecular Field (Weiss Field)
	13.2.Quantum Theory of Magnetism
		13.2.1.Spin Wave: Model of Localized Magnetism
		13.2.2.Heisenberg Hamiltonian
		13.2.3.X-Y Model
		13.2.4.Spin Waves in Ferromagnets
		13.2.5.Bosonization of Operators
		13.2.6.Magnetization
		13.2.7.Experiments Revealing Magnons
		13.2.8.Spin Waves in Antiferromagnets
		13.2.9.Bogoliubov Transformation
		13.2.10.Stability
		13.2.11.Spin Dynamics, Dynamical Response Function
			13.2.11.1.Spin Dynamics
		13.2.12.Response Function and Relaxation Time
			13.2.12.1.Linear Response Function
			13.2.12.2.The Fluctuation-Dissipation Theorem
			13.2.12.3.Onsager Relation
		13.2.13.Itinerant Ferromagnetism
			13.2.13.1.Quantum Impurities and the Kondo Effect
			13.2.13.2.Localized and Itinerant Spins Interaction
			13.2.13.3.Ruderman-Kittel-Kasuya-Yosida (RKKY) Interaction
			13.2.13.4.Abrikosov Technique
			13.2.13.5.Self-Energy of the Pseudo-Fermion
			13.2.13.6.Effective Spin Screening, Spin Susceptibility
			13.2.13.7.Second-Order Self-Energy Diagrams
			13.2.13.8.Scattering Amplitudes
			13.2.13.9.Scaling and Parquet Equation
			13.2.13.10.Kondo Effect and Numerical Renormalization Group
			13.2.13.11.Anisotropic Kondo Model
		13.2.14.Schwinger-Wigner Representation
		13.2.15.Jordan-Wigner
		13.2.16.Semi-Fermionic Representation: Hubbard Model
			13.2.16.1.Semi-Fermionic Representation
			13.2.16.2.Kondo Lattice: Effective Action
14.Nonequilibrium Quantum Field Theory
	14.1.Keldysh-Schwinger Technique: Time Contour
		14.1.1.Basic Features of the S-Matrix (Operator)
		14.1.2.Closed Time Path (CTP) Formalism
	14.2.Contour Green’s Functions
	14.3.Real-Time Formalism
		14.3.1.Real-Time Matrix Representation
	14.4.Two-Point Correlation Function Decomposition
	14.5.Equilibrium Green’s Function
		14.5.1.Spectral Function
			14.5.1.1.Kubo-Martin-Schwinger (KMS) Condition
		14.5.2.Sum Rule and Physical Interpretation
	14.6 Keldysh Rotation
	14.7 Path Integral Representation
		14.7.1.Gross-Pitaevskii Equation
	14.8.Dyson Equation and Self-Energy
	14.9.Nonequilibrium Generating Functional
	14.10.Gaussian Initial States
	14.11.Nonequilibrium 2PI Effective Action
		14.11.1.Luttinger-Ward Functional
	14.12 Kinetic Equation and the 2PI Effective Action
		14.12.1.The Self-Consistent Schwinger–Dyson Equation
	14.13.Closed Time Path (CTP) and Extended Keldysh Contours
	14.14.Kadanoff-Baym Contour
		14.14.1.Green’s Function on the Extended Contour
		14.14.2.Kadanoff-Baym Contour
	14.15.Kubo-Martin-Schwinger (KMS) Boundary Conditions
		14.15.1.Remark on KMS Boundary Conditions
		14.15.2.Generalization of an Average Value
	14.16.Neglect of Initial Correlations and Schwinger-Keldysh Limit
		14.16.1.Equation of Motion for the Nonequilibrium Green’s Function
			14.16.1.1.Nonequilibrium Green’s Function Equation of Motion: Auxiliary Fields and Functional Derivatives Technique
			14.16.1.2.Keldysh Initial Condition
			14.16.1.3.Perturbation Expansion and Feynman Diagrams
			14.16.1.4.Right- and Left-Hand Dyson Equations
			14.16.1.5.Self-Energy Self-Consistent Equations
	14.17.Kadanoff-Baym (KB) Formalism for Bose Superfluids
		14.17.1.Kadanoff-Baym Equations
			14.17.1.1.Fluctuation-Dissipation Theorem
			14.17.1.2.Wigner or Mixed Representation
	14.18.Green’s Function Wigner Transformation
References
Index