Crm Mathematical Physics

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 9 Conclusion 10 References A RG Flow and Operator Product Expansion 5 Disordered Quantum Solids T. Giamarchi and E. Orignac 1 Introduction 2 Disordered Interacting Fermions 2.1 Model 2.2 Pure System 2.3 Disorder 3 Tackling the Disorder 3.1 Chisel and Hammer 3.2 Starting from the Metal: RG 4 Other Systems and RG 5 A Zest of Numerics 6 Variational Method 6.1 A Classical Example 6.2 If it Ain't Broken 6.3 Quantum Problems 6.4 The Fine Prints 6.5 Higher Dimension: Electronic Crystals and Classical Systems 7 Commensurate Systems 7.1 The Peculiar Random Exchange 7.2 Mott versus Anderson 7.3 Variational Approach 7.4 Physical discussion 8 Conclusions 9 References III Functional Derivatives, Mean-Field, Self-Consistent Methods, Slave-Bosons and Extensions 6 Self-Consistent Many-Body Theory for Condensed Matter Systems N.E. Bickers 1 Introduction 2 Review of Mean-Field Theory 3 Basics of Functional Integration 3.1 Bose Systems 3.2 Fermi Systems 4 Self-Consistent Approximations for the Action Functional 5 O-Derivability and Thermodynamic Self-Consistency 6 Thermodynamic Derivatives 7 Crossing Symmetry 8 Parquet Equations 9 Spin Diagonalization 10 Fluctuation Exchange Approximation and Pseudopotential Parquet 11 Analysis of Ordering Instabilities 12 Renormalization Group Solution of SCF Equations 13 Some Numerical Examples 14 Conclusion 15 References 7 Fermi and Non-Fermi Liquid Behavior of Quantum Impurity Models: A Diagrammatic Pseudo-Particle Approach J. Kroha and P. Wol.e 1 Introduction 2 Single- and Multi-Channel Quantum Impurity Models 3 Pseudo-Particle Representation 3.1 Exact Projection Onto the Physical Hilbert Space 3.2 Analytical Properties and Infrared Behavior 4 Mean Field Approach and 1/N Expansion at 4.1 Slave Boson Mean Field Theory 4.2 1/N Expansion vs. Self-Consistent Formulation 5 Conserving Approximations: Gauge Invariant Self-Consistent Perturbation Theory in the Hybridization 5.1 Generating Functional 5.2 Noncrossing Approximation (NCA) 5.3 Evaluation of the Self-Consistency Equations at Low Temperatures 6 Conserving T-Matrix Approximation (CTMA) at 6.1 Dominant Contributions at Low Energy 6.2 Self-Consistent Formulation: CTMA 6.3 Results for the Auxiliary Particle Spectral Functions 6.4 Results for Physical Quantities: Spin Susceptibility 7 Anderson Model at Finite U: Generalized NCA and CTMA 7.1 Generating Functional 7.2 Results of SUNCA 8 Conclusion 9 References A Infrared Cancellation of Non-CTMA Diagrams A.1 Power Counting A.2 Infrared Cancellation 8 Conserving Approximations vs. Two-Particle Self-Consistent Approach S. Allen, A.-M.S. Tremblay, and Y.M. Vilk 1 Introduction 2 Functional Derivative Formalism and Conserving Approximations 2.1 Single-Particle Properties 2.2 Response Functions 2.3 Hartree-Fock and RPA as an Example 3 Another Approach 3.1 First Step: Two-Particle Self-Consistency for G(1), O(1), A(1) sp = Usp and A(1) ch = Uch 3.2 Second Step: An Improved Self-Energy O(2) 3.3 Internal Accuracy Check 4 Discussion and Extensions 5 References A An Approximate Formula for O(1)