Note: Cover -- Contents -- 1 Introduction -- 1.1 Continuous phase transitions and critical points -- 1.1.1 Divergences and critical exponents -- 1.1.2 Fluctuations and critical opalescence -- 1.2 The order parameter -- 1.2.1 Liquid-gas transition -- 1.2.2 Binary fluids -- 1.2.3 Ferromagnetic/paramagnetic transition -- 1.2.4 Anti-ferromagnetic/paramagnetic transition -- 1.2.5 Helium I/helium II transition -- 1.2.6 Conductor/superconductor transitions -- 1.2.7 Helium three -- 1.3 Correlation functions -- 1.4 Universality -- 1.5 Thermodynamic potentials -- 1.5.1 The Widom and Kadanoff scaling hypotheses -- 1.6 Why study phase transitions? -- Problems -- 2 Statistical mechanics -- 2.1 Thermodynamic quantities -- 2.2 Fluctuations and correlation functions -- 2.3 Metastability and spontaneous symmetry breaking -- 2.3.1 Metastability -- 2.3.2 Spontaneous symmetry breaking -- Problems -- 3 Models -- 3.1 Description of models -- 3.1.1 The Ising model -- 3.1.2 The lattice gas -- 3.1.3 ß-brass -- 3.1.4 The XY and Heisenberg models -- 3.1.5 Potts model -- 3.1.6 Gaussian and spherical models -- 3.1.7 Percolation model -- 3.2 Transfer matrices and the Ising ring -- 3.2.1 Solution of the Ising ring -- 3.2.2 Correlation functions -- 3.3 The partition function of the spherical model -- 3.4 High-temperature expansions and the Ising model -- 3.4.1 High-temperature expansions -- 3.4.2 The partition function of the Ising model -- 3.4.3 The correlation functions of the Ising model -- 3.4.4 Numerical evaluation of high-temperature expansions -- Problems -- 4 Numerical simulations -- 4.1 Direct evaluation of thermal averages -- 4.2 Sampling configurations -- 4.2.1 Importance sampling -- 4.2.2 General structure of numerical algorithms -- 4.3 Monte Carlo methods -- 4.3.1 The Metropolis algorithm -- 4.4 Molecular dynamics -- 4.4.1 Ergodicity and integrability. , 4.4.2 From microcanonical to canonical averages -- 4.5 Langevin equations -- 4.5.1 Comparison of the Langevin and molecular-dynamics methods -- 4.6 Independence of configurations -- 4.6.1 Correlations along the path -- 4.6.2 Critical slowing down -- 4.6.3 The Swendsen-Wang algorithm -- 4.6.4 The Wolff algorithm -- 4.7 Calculation of critical exponents from simulations -- Problems -- 5 Real-space renormalization -- 5.1 Renormalizing the lattice -- 5.2 Block variables -- 5.3 The renormalization of the Hamiltonian -- 5.3.1 Fixed points -- 5.3.2 The calculation of v -- 5.4 The renormalization of B, M, X and G[sub(c)] -- 5.4.1 The value of ω -- 5.4.2 Non-zero external field -- 5.4.3 The renormalization of M, χ and G[sub(c)] -- 5.4.4 Critical exponents for the renormalized model -- 5.5 The critical exponents for T = T[sub(c)] -- 5.5.1 The exponent η -- 5.5.2 The exponent δ -- 5.6 The critical exponents for T ≠ T[sub(c)] -- 5.6.1 The exponent β -- 5.6.2 The exponent γ -- 5.6.3 The exponent α -- 5.7 The scaling laws -- 5.8 Bond percolation in two dimensions -- 5.9 The Ising model -- 5.10 Monte Carlo renormalization -- Problems -- 6 Mean-field theory -- 6.1 Mean-field theory of the Ising model -- 6.2 Mean-field theory of percolation -- 6.3 Mean-field theory of the non-ideal gas -- 6.4 A variational derivation of mean-field theory -- 6.5 Correlation functions in mean-field theory -- 6.6 Infinite-range interactions -- 6.7 Critical exponents in mean-field theory -- 6.7.1 Calculating η from G[sup(2)][sub(c)](k) -- 6.8 What is missing from mean-field theory? -- Problems -- 7 The Landau-Ginzburg model -- 7.1 Formulation of the Landau-Ginzburg model -- 7.2 Landau theory -- Problems -- 8 Diagrammatic perturbation theory -- 8.1 The Gaussian partition function -- 8.1.1 Correlation functions in the Gaussian model. , 8.2 The partition function for the full Landau-Ginzburg model -- 8.2.1 The Feynman rules -- 8.2.2 The symmetry factor -- 8.3 The Helmholtz free energy of the Landau-Ginzburg model -- 8.3.1 Feynman rules in wavevector space -- 8.3.2 Vertex functions -- 8.4 The Gibbs free energy of the Landau-Ginzburg model -- 8.4.1 The rules for finding Γ[φ] -- 8.4.2 The loop expansion -- 8.4.3 The one-loop Gibbs free energy -- Problems -- 9 Renormalization -- 9.1 Mass renormalization -- 9.2 Field renormalization -- 9.3 Renormalizing the coupling constant -- 9.4 Renormalization at higher orders -- 9.5 More on field renormalization -- 9.6 The Ginzburg criterion -- Problems -- 10 The calculation of critical exponents for T ≥ Tc -- 10.1 Ultraviolet and infrared divergences -- 10.2 The calculation of γ -- 10.2.1 d = 4 and above -- 10.2.2 Below four dimensions -- 10.3 The calculation of η -- 10.3.1 d = 4 and above -- 10.3.2 Below four dimensions -- 10.4 The ε-expansion -- 10.4.1 Dimensional regularization -- 10.4.2 Calculating γ by dimensional regularization -- 10.4.3 Calculating η by dimensional regularization -- 10.4.4 Feynman parameters -- 10.4.5 The calculation of η again -- 10.4.6 Calculation of η by the ε-expansion -- Problems -- 11 The renormalization group -- 11.1 The renormalization group at T = T[sub(c)] -- 11.2 The exponents η and δ -- 11.2.1 The exponent η -- 11.2.2 The exponent δ -- 11.3 The calculation of β and γ[sub(1)] -- 11.3.1 The calculation of γ[sub(1)] to order ε[sup(2)] -- Problems -- 12 The renormalization group at T ≠ T[sub(c)] -- 12.1 Expansion about the critical temperature -- 12.1.1 Functional Taylor expansions -- 12.1.2 Diagrammatic representation of the Φ[sup(2)] correlation functions -- 12.1.3 Wavevector space -- 12.1.4 Vertex functions -- 12.1.5 Renormalization -- 12.1.6 Expanding the renormalized vertex functions. , 12.1.7 The validity of the expansion -- 12.2 The renormalization group equations -- 12.2.1 The exponent v -- 12.2.2 The exponent γ -- 12.2.3 The exponent α -- 12.3 The renormalization group below T[sub(c)] -- 12.3.1 The exponent β -- 12.4 Calculating γ[sub(2)] to one loop -- Problems -- 13 The lower critical dimension -- 13.1 Order below T[sub(c)] -- 13.1.1 The case D = 1 -- 13.1.2 Systems with more than one component -- 13.1.3 Goldstone modes -- 13.2 The non-linear σ-model -- 13.2.1 The two-point vertex function -- 13.2.2 The renormalization group equation -- 13.3 The Kosterlitz-Thouless transition -- 13.3.1 The two-dimensional Coulomb gas -- 13.3.2 General Remarks -- Problems -- 14 Universality -- 14.1 Perturbing the Gaussian Hamiltonian -- 14.1.1 The applicability of these results -- 14.2 Perturbing the Landau-Ginzburg Hamiltonian -- 14.2.1 The case of three dimensions -- 14.2.2 The case of two dimensions -- 14.3 Relevance and renormalizability -- Problems -- Appendices -- A: The magnetic scattering of neutrons -- B: The natural variables for thermodynamic potentials -- C: Magnetic energy -- D: Connected correlation functions and log Z[J] -- E: The Gibbs free energy -- F: Discrete Fourier transforms -- G: The method of steepest descent -- H: Counting closed loops on a square lattice -- I: Einstein's fluctuation theory -- J: The Gaussian transformation -- K: The Landau-Ginzburg model and the Ising model -- L: Functional differentiation and integration -- M: The Feynman rules for the vertex functions -- N: Feynman rules for generalized Landau-Ginzburg models -- Answers -- References -- Index -- A -- B -- C -- D -- E -- F -- G -- H -- I -- J -- K -- L -- M -- N -- O -- P -- Q -- R -- S -- T -- U -- V -- W -- X -- Z.