Lecture Notes On Field Theory In Condensed Matter Physics

Description

The aim of this book is to introduce a graduate student to selected concepts in condensed matter physics for which the language of field theory is ideally suited. The examples considered in this book are those of superfluidity for weakly interacting bosons, collinear magnetism, and superconductivity. Quantum phase transitions are also treated in the context of quantum dissipative junctions and interacting fermions constrained to one-dimensional position space. The style of presentation is sufficiently detailed and comprehensive that it only presumes familiarity with undergraduate physics. Preface Acknowledgments Contents Bosons 1. The harmonic crystal 1.1 Introduction 1.2 Classical one-dimensional crystal 1.3 Quantum one-dimensional crystal 1.4 Higher-dimensional generalizations 1.5 Problems 2. Bogoliubov theory of a dilute Bose gas Outline 2.1 Introduction 2.2 Second quantization for bosons 2.3 Bose-Einstein condensation and spontaneous symmetry breaking 2.4 Dilute Bose gas: Operator formalism at vanishing temperature 2.4.1 2.5 Dilute Bose gas: Path-integral formalism at any temperature 2.6 Problems 3. Non-Linear Sigma Models Outline 3.1 Introduction 3.2 Non-Linear Sigma Models (NLsM) 3.3 Fixed point theories, engineering and scaling dimensions, irrelevant, marginal, and relevant interactions 3.4 General method of renormalization 3.5 Perturbative expansion of the two-point correlation function up to one loop for the two-dimensional 3.6 Callan-Symanzik equation obeyed by the spin-spin correlator for the two-dimensional 3.7 Beta function for the 3.8 Problems 4. Kosterlitz-Thouless transition Outline 4.1 Introduction 4.2 Classical two-dimensional 4.3 The Coulomb-gas representation of the classical twodimensional 4.4 Equivalence between the Coulomb gas and Sine-Gordon model 4.5 Fugacity expansion of n-point functions in the Sine-Gordon model 4.6 Kosterlitz renormalization-group equations 4.7 Problems Fermions 5. Non-interacting fermions Outline 5.1 Introduction 5.2 Second quantization for fermions 5.3 The non-interacting jellium model 5.4 Time-ordered Green functions 5.5 Problems 6. Jellium model for electrons in a solid Outline 6.1 Introduction 6.2 Definition of the Coulomb gas in the Schr¨odinger picture 6.3 Path-integral representation of the Coulomb gas 6.4 The random-phase approximation 6.5 Diagrammatic interpretation of the random-phase approximation 6.6 Ground-state energy in the random-phase approximation 6.7 Lindhard response function 6.8 Random-phase approximation for a short-range interaction 6.9 Feedback effect on and by phonons 6.10 Problems 7. Superconductivity in the mean-field and random-phase approximations Outline 7.1 Pairing-order parameter 7.2 Scaling of electronic interactions 7.2.1 7.3 Timeand space-independent Landau-Ginzburg action 7.4 Mean-field theory of superconductivity 7.5 Nambu-Gorkov representation 7.6 Effective action for the pairing-order parameter 7.7 Effective theory in the vicinity of 7.8 Effective theory in the vicinity of 7.9 Problems 8. A single dissipative Josephson junction Outline 8.1 Phenomenological model of a Josephson junction 8.2 DC-Josephson effect 8.3 AC-Josephson effect 8.4 Dissipative Josephson junction 8.5 Instantons in quantum mechanics 8.6 The quantum-dissipative Josephson junction 8.7 Duality in a dissipative Josephson junction 8.8 Renormalization-group methods 8.9 Conjectured phase diagram for a dissipative Josephson junction 8.10 Problems 9. Abelian bosonization in two-dimensional space and time Outline 9.1 Introduction 9.2 Abelian bosonization of the Thirring model 9.3 Applications 9.4 Problems Appendix A The harmonic-oscillator algebra and its coherent states A.1 The harmonic-oscillator algebra and its coherent states A.1.1 A.2 Path-integral representation of the anharmonic oscillator A.3 Higher dimensional generalizations Appendix B Some Gaussian integrals B.1 Generating function B.2 Bose-Einstein distribution and the residue theorem Appendix C Non-Linear Sigma Models (NLσM) on Riemannian manifolds C.1 Introduction C.2 A few preliminary definitions C.3 Definition of a NLsM on a Riemannian manifold C.4 Classical equations of motion for NLsM: Christoffel symbol and geodesics C.5 Riemann, Ricci, and scalar curvature tensors C.6 Normal coordinates and vielbeins for NLsM C.6.1 C.7 How many couplings flow on a NLsM? Appendix D The Villain model Appendix E Coherent states for fermions, Jordan-Wigner fermions, and linear-response theory E.1 Grassmann coherent states E.2 Path-integral representation for fermions E.3 Jordan-Wigner fermions E.4 The ground state energy and the single-particle time-ordered Green function E.5 Linear response Appendix F Landau theory of Fermi liquids Introduction F.1 Adiabatic continuity F.2 Quasiparticles F.3 Topological stability of the Fermi surface F.4 Quasiparticles in the Landau theory of Fermi liquids as poles of the two-point Green function F.5 Breakdown of Landau Fermi liquid theory F.5.1 Appendix G First-order phase transitions induced by thermal fluctuations Outline G.1 Landau-Ginzburg theory and the mean-field theory of continuous phase transitions G.2 Fluctuations induced by a local gauge symmetry G.3 Applications Appendix H Useful identities H.1 Proof of Equation (8.75) Appendix I Non-Abelian bosonization I.1 Introduction I.2 Minkowski versus Euclidean spaces I.3 Free massless Dirac fermions and the Wess-Zumino-Witten theory I.4 A quantum-mechanical example of a Wess-Zumino action I.5 Wess-Zumino action in (1+1)-dimensional Minkowski space and time I.6 Equations of motion for the WZNW action I.7 One-loop RG flow for the WZNW theory I.8 The Polyakov-Wiegmann identity I.9 Integration of the anomaly in QCD I.10 Bosonization of QCD Bibliography Index