Real-Space Renormalization

1. Progress and Problems in Real-Space Renormalization.- 1.1 Introduction.- 1.2 Review of Real-Space Renormalization.- 1.3 New Renormalization Methods.- 1.3.1 Bond-Moving and Variational Methods.- 1.3.2 Monte Carlo Renormalization.- 1.3.3 Exact Differential Transformations.- 1.3.4 Phenomenological Renormalization.- 1.4 New Applications.- 1.4.1 Adsorbed Systems.- 1.4.2 Applications to Quantum Systems.- 1.4.3 Percolation and Polymers.- 1.4.4 Dynamic Real-Space Renormalization.- 1.4.5 The Kosterlitz-Thouless Transition.- 1.4.6 Field-Theoretical Applications.- 1.5 Fundamental Problems.- 1.5.1 Choice of the Weight Function.- 1.5.2 Griffiths-Pearce Peculiarities.- 1.6 Exact Differential Real-Space Renormalization.- 1.6.1 The Two-Dimensional Ising Model.- 1.6.2 Discussion.- 1.7 Phenomenological Renormalization.- 1.7.1 Description of the Method.- 1.7.2 Applications.- 1.8 Concluding Remarks.- References.- 2. Bond-Moving and Variational Methods in Real-Space Renormalization.- 2.1 Introduction.- 2.2 Variational Principles.- 2.2.1 Lower-Bound Property of Bond-Moving Approximations.- 2.2.2 Upper-Bound Property of the First-Order Cumulant Approximation.- 2.3 The Migdal-Kadanoff Transformation.- 2.3.1 Application to the Ising Model with Nearest-Neighbor Interactions.- 2.3.2 Inclusion of a Magnetic Field.- 2.3.3 The Bond-Moving Prescription of EMERY and SWENDSEN.- 2.3.4 Inconsistent Scaling of the Correlation Function.- 2.3.5 Relation to Exactly Soluble Hierarchical Models.- 2.3.6 Applications.- 2.3.7 Modifications of the Migdal-Kadanoff Procedure.- 2.4 Variational Transformations.- 2.4.1 The Kadanoff'Lower-Bound Variational Transformation.- 2.4.2 The Kadanoff Criterion for the Optimal Variational Parameter.- 2.4.3 Problems with the Lower-Bound Variational Transformation.- 2.4.4 Determination of an Optimal Sequence of Variational Parameters.- 2.4.5 Applications of the Lower-Bound Variational Transformation.- 2.4.6 Other Variational Methods.- 2.5 Conclusion.- References.- 3. Monte Carlo Renormalization.- 3.1 Introduction.- 3.2 Basic Notation and Renormalization-Group Formalism.- 3.3 Large-Cell Monte Carlo Renormalization Group.- 3.4 MCRG.- 3.4.1 Calculation of Critical Exponents.- 3.4.2 Calculation of Renormalized Coupling Constants.- 3.5 MCRG Calculations for Specific Systems.- 3.6 Other Approaches to the Monte Carlo Renormalization Group.- 3.7 Conclusions.- References.- 4. The Real-Space Dynamic Renormalization Group.- 4.1 Introduction.- 4.2 Dynamic Problem of Interest.- 4.3 RSDRG - Formal Development.- 4.4 Implementation of the RSDRG Using Perturbation Theory.- 4.4.1 General Development.- 4.4.2 Expansion for H and D?.- 4.4.3 Solution to the Zeroth-Order Problem.- 4.4.4 Renormalization to First Order.- 4.4.5 Recursion Relations for the Correlation Functions.- 4.5 Determination of Parameters.- 4.5.1 General Comments.- 4.5.2 The Parameters K0 and KR0.- 4.5.3 The Dynamic Parameters ?0 and ?.- 4.6 Results.- 4.7 Discussion.- References.- 5. Renormalization for Quantum Systems.- 5.1 Background.- 5.2 Application of the Niemeijer-van Leeuwen Renormalization Group Method to Quantum Lattice Models.- 5.3 The Block Method.- 5.3.1 Principles.- 5.3.2 Applications.- a) The Ising Model in a Transverse Field in One Dimension.- b) The Free Fermion Model in One Dimension.- 5.3.3 Extensions of the Method.- a) Extension to Large Blocks.- b) Extension by Increasing the Number nL of Levels Retained.- c) Other Extensions.- 5.4 Applications of the Block Method.- 5.4.1 Spin Systems.- a) The Spin 1/2 Ising Model in a Transverse Field (ITF).- b) The XY Heisenberg Spin 1/2 Chain.- c) The XY Model in a Z Field for d = 2, 3.- d) The Spin 1 XY Model with an Anisotropy Field for d = 1.- 5.4.2 Fermion Systems.- a) The d = 1 Hubbard Model.- b) Interacting Fermions in d = 1.- c) One-Dimensional Model of f and d Electrons with Hybridization V and fd Interaction Ufd.- 5.4.3 Spin Fermion Systems: The Kondo Lattice in d = 1.- 5.4.4 Quantum Versions of Classical Statistical Mechanics in 1 + 1 Dimension.- a) The 0(n) Model.- b) The P(q) Potts Model.- c) Tricritical Point for Ising Systems in 1 + 1 Dimensions.- 5.4.5 Applications to Field Theory.- a) The Thirring Model in One-Space and One-Time Dimension.- b) The U(1) Goldstone Model in Two Dimensions.- c) Lattice Gauge Theories.- 5.5 Discussion.- 5.5.1 When is the BRG More Suitable?.- 5.5.2 How to Control the Method?.- a) The Division of the Lattice into Blocks.- b) Which Levels to Retain for the Truncated Basis?.- 5.5.3 What Has Been Done and What Are the Difficulties Encountered?.- a) Quantum Properties at T = 0.- b) Quantum Properties at T ? 0.- c) Difficulties.- 5.5.4 Comparison Between Different Methods.- a) The Real-Space RG Methods for Classical Systems.- b) Finite-Size Scaling Methods.- 5.6 What to Do Next?.- 5.6.1 Improvement of the Method.- 5.6.2 Applications.- References.- 6. Application of the Real-Space Renormalization to Adsorbed Systems.- 6.1 Introduction.- 6.2 The Sublattice Method.- 6.3 The Prefacing Method and Introduction of Vacancies.- 6.4 The Potts Model.- 6.5 Further Applications of the Vacancy.- 6.6 Summary.- References.- 7. Position-Space Renormalization Group for Models of Linear Polymers, Branched Polymers, and Gels.- 7.1 Three Physical Systems.- 7.1.1 Linear Polymers.- 7.1.2 Branched Polymers.- 7.1.3 Gels.- 7.2 Three Mathematical Models.- 7.2.1 Percolation.- 7.2.2 Self-Avoiding Walks.- 7.2.3 Lattice Animals.- 7.3 Position-Space Renormalization Group Treatment.- 7.3.1 Percolation.- a) Basic Approach.- b) Extensions.- 7.3.2 Self-Avoiding Walks.- a) Basic Approach.- b) Extensions.- 7.3.3 Lattice Animals.- a) Basic Approach.- b) Extensions.- 7.4 Other Approaches.- 7.4.1 Percolation.- 7.4.2 Self-Avoiding Walks.- 7.4.3 Lattice Animals.- 7.5 Concluding Remarks and Outlook.- References.