Theoretical Methods for Strongly Correlated Electrons

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