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Contents Series Preface Preface C. Bourbonnais, D. Senechal, A. Ruckenstein, and A.-M.S. Tremblay I Numerical Methods 1 Density Matrix Renormalization Karen Hallberg 1 Introduction 2 The Method 3 Applications 4 Other Extensions to DMRG 4.1 Classical Systems 4.2 Finite-Temperature DMRG 4.3 Phonons, Bosons and Disorder 4.4 Molecules and Quantum Chemistry 5 Dynamical Correlation Functions 5.1 Lanczos and Correction Vector Techniques 5.2 Moment Expansion 5.3 Finite Temperature Dynamics 6 Conclusions 7 References 2 Quantum Monte Carlo Methods for Strongly Correlated Electron Systems Shiwei Zhang 1 Introduction 2 Preliminaries 2.1 Starting Point of Quantum Monte Carlo (QMC) 2.2 Basics of Monte Carlo Techniques 2.3 Slater Determinant Space 2.4 Hubbard-Stratonovich Transformation 3 Standard Auxiliary-Field Quantum Monte Carlo 3.1 Ground-State Method 3.2 Finite-Temperature Method 4 Constrained Path Monte Carlo Methods-Ground-State and Finite-Temperature 4.1 Why and How Does the Sign Problem Occur? 4.2 The Constrained-Path Approximation 4.3 Ground-State Constrained Path Monte Carlo (CPMC) Method 4.4 Finite-Temperature Method 4.5 Additional Technical Issues 5 Illustrative Results 6 Summary 7 References A Brief Review of Con.guration-Space Methods A.1 Variational Monte Carlo A.2 Green's Function Monte Carlo (GFMC) II Lagrangian, Functional Integral, Renormalization Group, Conformal and Bosonization Methods Renormalization Group Technique for Quasi-One-Dimensional Interacting Fermion Systems at Finite Temperature C. Bourbonnais, B. Guay and R. Wortis 1 Introduction 2 Scaling Ansatz for Fermions 2.1 One Dimension 2.2 Anisotropic Scaling and Crossover Phenomena 3 Free Fermion Limit 3.1 One Dimension 3.2 Interchain Coupling 4 The Kadano.-Wilson Renormalization Group 4.1 One-Dimensional Case 4.2 One-Loop Results 4.3 Two-Loop Results 4.4 Response Functions 5 Interchain Coupling: One-Particle Hopping 5.1 Interchain Pair Hopping and Long-Range Order 5.2 Long-Range Order in the Decon.ned Region 6 Kohn-Luttinger Mechanism in Quasi-One-Dimensional Metals 6.1 Generation of Interchain Pairing Channels 6.2 Possibility of Long-Range Order in the Interchain Pairing Channels 7 Summary and Concluding Remarks 8 References A One-Particle Self-Energy at the Two-Loop Level A.1 Backward and Forward Scattering Contributions A.2 Umklapp contribution 4 An Introduction to Bosonization D. Senechal 1 Quantum Field Theory in Condensed Matter 2 A Word on Conformal Symmetry 2.1 Scale and Conformal Invariance 2.2 Conformal Transformations 2.3 E.ect of Perturbations 2.4 The Central Charge 3 Interacting Electrons in One Dimension 3.1 Continuum Fields and Densities 3.2 Interactions 4 Bosonization: A Heuristic View 4.1 Why Is One-Dimension Special? 4.2 The Simple Boson 4.3 Bose Representation of the Fermion Field 5 Details of the Bosonization Procedure 5.1 Left and Right Boson Modes 5.2 Proof of the Bosonization Formulas: Vertex Operators 5.3 Bosonization of the Free-Electron Hamiltonian 5.4 Spectral Equivalence of Boson and Fermion 5.5 Case of Many Fermion Species: Klein Factors 5.6 Bosonization of Interactions 6 Exact Solution of the Tomonaga-Luttinger Model 6.1 Field and Velocity Renormalization 6.2 Left-Right Mixing 6.3 Correlation Functions 6.4 Spin or Charge Gap 7 Non-Abelian Bosonization 7.1 Symmetry Currents 7.2 Application to the Perturbed Tomonaga-Luttinger Model 8 Other Applications of Bosonization 8.1 The Spin- 12 Heisenberg Chain 8.2 Edge States in Quantum Hall Systems 8.3 And More