Condensed Matter Field Theory

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Contents
Preface
1 From particles to fields
1.1 Classical harmonic chain: phonons
Lagrangian formulation and equations of motion
Hamiltonian formulation
1.2 Functional analysis and variational principles
1.3 Maxwell’s equations as a variational principle
1.4 Quantum chain
Revision of the quantum harmonic oscillator
Quasi-particle interpretation of the quantum chain
1.5 Quantum electrodynamics
Field quantization
Vacuum fluctuations in matter
1.6 Noether’s theorem
Symmetry transformations
Example: translational invariance
1.7 Summary and outlook
1.8 Problems
Electrodynamics from a variational principle
Hamiltonian of electromagnetic field
Phonon specific heat
Van der Waals force
2 Second quantization
2.1 Introduction to second quantization
Motivation
The apparatus of second quantization
Occupation number representation and Fock space
Foundations of second quantization
Practical aspects
2.2 Applications of second quantization
Electrons in a periodic potential
Nearly free electron systems
Tight–binding systems
Interaction effects in the tight-binding system
Mott–Hubbard transition and the magnetic state
Interacting fermions in one dimension
Qualitative discussion
Quantitative analysis
Quantum spin chains
Quantum ferromagnet
Quantum antiferromagnet
2.3 Summary and outlook
2.4 Problems
Stone–von Neumann theorem
Semiclassical spin waves
Su–Shrieffer–Heeger model of a conducting polymer chain
Schwinger boson representation
Jordan–Wigner transformation
Spin–charge separation in one-dimension
The Kondo problem
3 Feynman path integral
3.1 The path integral: general formalism
3.2 Construction of the path integral
Path integral and statistical mechanics
Semiclassics from the path integral
Construction recipe of the path integral
3.3 Applications of the Feynman path integral
Quantum particle in a well
Double well potential: tunneling and instantons
The instanton gas
Escape from a metastable minimum: “bounces”
Tunneling of quantum fields: “fate of the false vacuum”
Tunneling in a dissipative environment
Caldeira–Leggett model
Disssipative quantum tunneling
Path integral for spin
A reminder of finite-dimensional SU(2)-representation theory
Construction of the path integral
Analysis of the action
Trace formulae and quantum chaos
Semiclassical approximation to the density of states
3.4 Summary and outlook
3.5 Problems
Quantum harmonic oscillator
Density matrix
Depinning transition and bubble nucleation
Tunneling in a dissipative environment
Winding numbers
Particle in a periodic potential
4 Functional field integral
4.1 Construction of the many-body path integral
Coherent states (bosons)
Coherent states (fermions)
4.2 Field integral for the quantum partition function
Partition function of non-interacting gas
4.3 Field theoretical bosonization: a case study
One-dimensional electron gas (fermionic theory)
Non-interacting system
Interacting case
One-dimensional electron gas (bosonic theory)
Non-interacting system
Interacting system
4.4 Summary and outlook
4.5 Problems
Exercises on fermion coherent states
Feynman path integral from the functional field integral
Quantum partition function of the harmonic oscillator
Boson–fermion duality
Frequency summations
Pauli paramagnetism
Electron–phonon coupling
Disordered quantum wires
5 Perturbation theory
5.1 General structures and low-order expansions
An instructive integral
4-theory
Perturbation theory at low orders
5.2 Ground state energy of the interacting electron gas
Qualitative aspects
Perturbative approach
First-order perturbation theory
Second-order perturbation theory
Higher orders in perturbation theory
5.3 Infinite-order expansions
Self-energy operator
Large-N expansion
5.4 Summary and outlook
5.5 Problems
Technical aspects of diagrammatic perturbation theory
Self-consistent T-matrix approximation
Kondo effect: perturbation theory
6 Broken symmetry and collective phenomena
6.1 Mean-field theory
6.2 Plasma theory of the interacting electron gas
6.3 Bose–Einstein condensation and super fluidity
Bose–Einstein condensation
The weakly interacting Bose gas
Spontaneous symmetry breaking
Superfluidity
6.4 Superconductivity
Basic concepts of BCS theory
Cooper instability
Mean-field theory of superconductivity
Ground state
Excitations
Superconductivity from the field integral
Mean-field theory
Ginzburg–Landau theory
Action of the Goldstone mode
Meissner effect and Anderson–Higgs mechanism
6.5 Field theory of the disordered electron gas
Disorder in metals
Replica field theory
Basic notions of impurity scattering
Diffusion
Mean-field theory and spontaneous symmetry breaking
Low-energy field theory
6.6 Summary and outlook
6.7 Problems
Peierls instability
Temperature profile of the BCS gap
Fluctuation contribution to the Ginzburg–Landau action of the superconductor
Coulomb blockade
Action of a tunnel junction
Josephson junction
Field theory of the BCS to BEC crossover
Metallic magnetism
Functional bosonization
7 Response functions
7.1 Crash course in modern experimental techniques
7.1.1 Basic concepts
7.1.2 Experimental methods
Thermodynamic experiments
Transport experiments
Spectroscopic experiments
Other experimental techniques
7.2 Linear response theory
7.2.1 Microscopic response theory
7.3 Analytic structure of correlation functions
7.3.1 Sum rules and other exact identities
The spectral (density) function
The dielectric function: a case study
Experimental access to the spectral density function
7.4 Electromagnetic linear response
Electromagnetic response of the microscopic theory
Electromagnetic response of effective theories
7.4.1 Longitudinal conductivity of the disordered electron gas
7.5 Summary and outlook
7.6 Problems
7.6.1 Orthogonality catastrophe
7.6.2 RPA dielectric function
7.6.3 Electromagnetic response of a quantum dot
7.6.4 Hall conductivity
8 The renormalization group
8.1 The one-dimensional Ising model
8.1.1 Exact solution
8.1.2 Elements of scaling theory
8.1.3 Kadanoff’s block spin RG
8.2 Dissipative quantum tunneling
8.3 Renormalization group: general theory
8.3.1 Gell-Mann–Low equations
I: Subdivision of the field manifold
II: RG step
III: Rescaling
8.3.2 Analysis of the Gell-Mann–Low equation
8.3.3 Scaling theory
Scaling functions
Scaling functions and critical exponents
8.4 RG analysis of the ferromagnetic transition
8.4.1 Preliminary dimensional analysis
8.4.2 Landau mean-field theory
8.4.3 Gaussian model
8.4.4 Renormalization group analysis
Step I
Steps II and III
8.5 RG analysis of the nonlinear σ-model
8.5.1 Field integrals over groups
8.5.2 One-loop expansion
8.6 Berezinskii–Kosterlitz–Thouless transition
8.6.1 Vortices and the topological phase transition
8.6.2 RG analysis of the BKT transition
8.7 Summary and outlook
8.8 Problems
8.8.1 Dissipative quantum tunneling: strong potential limit
8.8.2 Quantum criticality
8.8.3 RG analysis of the nonlinear σ-model II
8.8.4 Scaling theory of the Anderson metal insulator transition
8.8.5 Kondo effect: poor man’s scaling
9 Topology
9.1 Example: particle on a ring
9.2 Homotopy
9.2.1 Generalities
9.2.2 Examples of homotopies
9.3 θ-Terms
9.3.1 A case study…
9.3.2 Functional integration and topological textures: generalities
9.3.3 Spin chains
9.3.4 Integer quantum Hall effect
9.3.5 Background information on the IQHE
9.3.6 IQHE as a topological phenomenon
9.3.7 Field theory of the integer quantum Hall effect
Pruisken’s field theory: construction
Pruisken’s field theory: long-range physics
Quantum Hall transition
9.4 Wess–Zumino terms
9.4.1 A crash-course in differential geometry
Coordinate representations
Tangent space
Differential forms
Integration on manifolds
9.4.2 From θ- to Wess–Zumino terms
The geometry of θ-terms
The geometry of Wess–Zumino terms
9.4.3 Example: magnetic moment coupled to fermions
9.4.4 Spin chains: beyond the semi–classical limit
Fermion representation of the antiferromagnetic spin chain
Non-abelian bosonization
Renormalization group flow of the WZW model
WZW model of interacting fermions
9.5 Chern–Simons terms
9.5.1 Fractional quantum Hall effect (FQHE)
9.5.2 Chern–Simons field theory: construction
Singular gauge transformation
Derivation of the Chern–Simons action
Particle exchange in two dimensions
9.5.3 Chern–Simons field theory II: analysis
Mean-field equations
Fluctuations
9.6 Summary and outlook
9.7 Problems
9.7.1 Persistent current of a disordered ring
9.7.2 Working with the SU(N) Wess–Zumino term
9.7.3 Renormalization group analysis of the SU(N) Wess–Zumino model
9.7.4 Fractional quantum Hall effect: physics at the edge
10 Nonequilibrium (classical)
10.1 Fundamental questions of (nonequilibrium) statistical mechanics
10.2.1 Fluctuation–Dissipation Theorem (FDT)
10.2.2 A brief compendium on noise
Johnson–Nyquist noise
Shot noise
Other sources of noise
10.2.3 Fokker–Planck equation I
10.2.4 Beyond equilibrium
Active Brownian motion
Swarms
10.3 Boltzmann kinetic theory
10.3.1 Derivation of the Boltzmann equation
10.3.2 Discussion of the Boltzmann equation
The Boltzmann H-Theorem
Mesoscopic evolution laws
Beyond equilibrium: zero modes of the collision integral
Example: thermal conductivity of a gas of particles
10.4 Stochastic processes
10.4.1 The notion of a stochastic process
10.4.2 Markov processes
Chapman–Kolmogorov relation and master equation
Example: Gaussian process
Example: Poisson process
10.4.3 Fokker–Planck equation II
10.4.4 Quality of the Fokker–Planck approximation: an example
10.5 Field theory I: zero dimensional theories
10.5.1 Martin–Siggia–Rose–Janssen–de Dominicis approach
10.5.2 Field integral representation of the master equation I
10.5.3 Doi–Peliti operator technique
10.6 Field theory II: higher dimensions
10.6.1 Basic notions of dynamical critical phenomena
10.6.2 Field theories of finite dimensional Langevin systems
10.6.3 Field theory of finite dimensional stochastic processes
10.6.4 Fluctuation–dissipation theorem (revisited)
FDT I: Equilibrium linear response
FDT II: linear Langevin equations
FDT III: MSRJD field theory
10.7 Field theory III: applications
10.7.1 Driven diffusive lattice gases
Microscopic formulation
Mesoscopic formulation
Above criticality: consequences of FDT violation
The system at criticality
Perturbative RG
10.7.2 Directed percolation
Directed Percolation: Phenomenology
Elements of scaling theory
Field theory
Perturbative RG
10.8 Summary and Outlook
10.9 Problems
10.9.1 Wigner surmise
10.9.2 Ornstein-Uhlenbeck process
10.9.3 Ornstein-Uhlenbeck process revisited
10.9.4 Directed percolation
11 Nonequilibrium (quantum)
11.1 Prelude: Quantum master equation
11.1.1 Derivation of the master equation
11.1.2 Example: oscillator coupled to a bath
11.2 Keldysh formalism: basics
11.2.1 The idea
11.2.2 Case study
11.2.3 Continuum field theory
11.2.4 Generalization
Retarded and advanced Green function
Keldysh Green function
Interaction
11.2.5 Fluctuation dissipation theorem
11.2.6 Classical limit I
11.3 Particle coupled to an environment
Keldysh theory of a quantum particle
Coupling to an oscillator bath
Integration over oscillator modes
Langevin equation
11.4 Fermion Keldysh theory (a list of changes)
11.4.1 Single level
11.4.2 Generalization
11.5 Kinetic equation
11.5.1 Quasiclassical theory
Wigner transform
Derivation of the kinetic equation
Collision term
11.6 A mesoscopic application
11.6.1 Out-of-equilibrium quantum dot
11.6.2 Dot distribution function
Trial dot distribution function
Tunneling action
11.6.3 Observables
11.6.4 Open quantum dot
Classical resistor network
Zero bias anomaly
11.7 Full counting statistics
11.7.1 Generalities
11.7.2 Realizations of current noise
11.7.3 Full counting statistics of the double barrier quantum dot
11.7.4 General ramifications of FCS
11.8 Summary and outlook
11.9 Problems
11.9.1 Atom-field Hamiltonian
11.9.2 Atom-field Hamiltonian II: Weisskopf–Wigner theory of spontaneous
11.9.3 Keldysh theory of the Coulomb blockade
Index