Real-Space Renormalization
Published on 1. July 1982
Book
Hardback
XIV, 216 pages
978-3-540-11459-8 (ISBN)
Description
The renormalization-group approach is largely responsible for the considerable success which has been achieved in the last ten years in developing a complete quantitative theory of phase transitions. Before, there was a useful physical picture of phase transitions, but a general method for making accurate quantitative predictions was lacking. Existent theories, such as the mean-field theory of Landau, sometimes reproduce phase diagrams reliably but were known to fail qualitatively near critical points, where the critical behavior is particularly interesting be cause of its universal character. In the mid 1960's Widom found that the singularities in thermodynamic quanti ties were well described by homogeneous functions. Kadanoff extended the homogeneity hypothesis to correlation functions and linked it to the idea of scale invariance. In the early 1970's Wilson showed how Kadanoff's rescaling could be explicitly carried out near the fixed point of a flow in Hamiltonian space. He made the first practical renormalization-group calculation of the flow induced by the elimination of short-wave-length Fourier components of the order-parameter field. The univer sality of the critical behavior emerges in a natural way in this approach, with a different fixed point for each universality class. The discovery by Wilson and Fisher of a systematic expansion procedure in E for a system in d = 4 - E dimen sions was followed by a cascade of calculations of critical quantities as a function of d and of the order-parameter dimensionality n.
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Content
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.