Condensed Matter Field Theory

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Corrected digital edition. No missing equations or text. Half-title Title Copyright 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