- Title Pages
- Preface
- Acknowledgments
- General References
- 1 Algebraic Preliminaries
- 2 Euclidean Path Integrals In Quantum Mechanics
- 3 Path Integrals In Quantum Mechanics: Generalizations
- 4 Stochastic Differential Equations: Langevin, Fokker–Planck Equations
- 5 Path And Functional Integrals In Quantum Statistical Physics
- 6 Quantum Evolution: From Particles To Fields
- 7 Quantum Field Theory: Functional Methods and Perturbation Theory
- 8 Relativistic Fermions
- 9 Quantum Field Theory: Divergences and Regularization
- 10 Introduction to Renormalization Theory. Renormalization Group Equations
- 11 Dimensional Regularization, Minimal Subtraction: RG Functions
- 12 Renormalization Of Composite Operators. Short Distance Expansion
- 13 Symmetries And Renormalization
- 14 The Non-Linear σ-Model: An Example Of a Non-Linear Symmetry
- 15 General Non-Linear Models In Two Dimensions
- 16 St And Brs Symmetries, Stochastic Field Equations
- 17 From Langevin Equation To Supersymmetry
- 18 Abelian Gauge Theories
- 19 Non-Abelian Gauge Theories: Introduction
- 20 The Standard Model. Anomalies
- 21 Gauge Theories: Master Equation And Renormalization
- 22 Classical And Quantum Gravity. Riemannian Manifolds And Tensors
- 23 Critical Phenomena: General Considerations
- 24 Mean Field Theory For Ferromagnetic Systems
- 25 General Renormalization Group. The Critical Theory Near Dimension Four
- 26 Scaling Behaviour In The Critical Domain
- 27 Corrections to Scaling Behaviour
- 28 Non-Magnetic Systems and The (φ<sup>2</sup>)<sup>2</sup> Field Theory
- 29 Calculation Of Universal Quantities
- 30 The <i>O</i>(<i>N</i>) Vector Model For <i>N</i> Large
- 31 Phase Transitions Near Two Dimensions
- 32 Two-Dimensional Models and Bosonization Method
- 33 The <i>O</i>(2) Classical Spin Model In Two Dimensions
- 34 Critical Properties Of Gauge Theories
- 35 Uv Fixed Points In Quantum Field Theory
- 36 Critical Dynamics
- 37 Field Theory in a Finite Geometry: Finite Size Scaling
- 38 Quantum Field Theory At Finite Temperature: Equilibrium Properties
- 39 Instantons In Quantum Mechanics
- 40 Unstable Vacua In Quantum Field Theory
- 41 Degenerate Classical Minima And Instantons
- 42 Perturbation Series At Large Orders. Summation Methods
- 43 Multi-Instantons In Quantum Mechanics
- Index
Scaling Behaviour In The Critical Domain
Scaling Behaviour In The Critical Domain
- Chapter:
- (p.636) 26 SCALING BEHAVIOUR IN THE CRITICAL DOMAIN
- Source:
- Quantum Field Theory and Critical Phenomena
- Author(s):
JEAN ZINN-JUSTIN
- Publisher:
- Oxford University Press
Chapter 25 established the scaling behaviour of correlation functions at criticality, T = Tc . This chapter studies the critical domain which is defined by the property that the correlation length is large with respect to the microscopic scale, but finite. Using results proven in Chapter 10, it first demonstrates strong scaling above Tc : in the critical domain above Tc , all correlation functions, after rescaling, can be expressed in terms of universal correlation functions, in which the scale of distance is provided by the correlation length. The first part of the chapter is restricted to Ising-like systems. It then generalizes the results to N-component order parameters in Section 26.6. In Section 26.7, shows how to expand the universal two-point function when T approaches Tc , using the short distance expansion. Finally, the appendix discusses the energy correlation function when the specific heat exponent vanishes.
Keywords: correlation functions, Ising-like systems, energy correlation, specific heat exponent
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- Title Pages
- Preface
- Acknowledgments
- General References
- 1 Algebraic Preliminaries
- 2 Euclidean Path Integrals In Quantum Mechanics
- 3 Path Integrals In Quantum Mechanics: Generalizations
- 4 Stochastic Differential Equations: Langevin, Fokker–Planck Equations
- 5 Path And Functional Integrals In Quantum Statistical Physics
- 6 Quantum Evolution: From Particles To Fields
- 7 Quantum Field Theory: Functional Methods and Perturbation Theory
- 8 Relativistic Fermions
- 9 Quantum Field Theory: Divergences and Regularization
- 10 Introduction to Renormalization Theory. Renormalization Group Equations
- 11 Dimensional Regularization, Minimal Subtraction: RG Functions
- 12 Renormalization Of Composite Operators. Short Distance Expansion
- 13 Symmetries And Renormalization
- 14 The Non-Linear σ-Model: An Example Of a Non-Linear Symmetry
- 15 General Non-Linear Models In Two Dimensions
- 16 St And Brs Symmetries, Stochastic Field Equations
- 17 From Langevin Equation To Supersymmetry
- 18 Abelian Gauge Theories
- 19 Non-Abelian Gauge Theories: Introduction
- 20 The Standard Model. Anomalies
- 21 Gauge Theories: Master Equation And Renormalization
- 22 Classical And Quantum Gravity. Riemannian Manifolds And Tensors
- 23 Critical Phenomena: General Considerations
- 24 Mean Field Theory For Ferromagnetic Systems
- 25 General Renormalization Group. The Critical Theory Near Dimension Four
- 26 Scaling Behaviour In The Critical Domain
- 27 Corrections to Scaling Behaviour
- 28 Non-Magnetic Systems and The (φ<sup>2</sup>)<sup>2</sup> Field Theory
- 29 Calculation Of Universal Quantities
- 30 The <i>O</i>(<i>N</i>) Vector Model For <i>N</i> Large
- 31 Phase Transitions Near Two Dimensions
- 32 Two-Dimensional Models and Bosonization Method
- 33 The <i>O</i>(2) Classical Spin Model In Two Dimensions
- 34 Critical Properties Of Gauge Theories
- 35 Uv Fixed Points In Quantum Field Theory
- 36 Critical Dynamics
- 37 Field Theory in a Finite Geometry: Finite Size Scaling
- 38 Quantum Field Theory At Finite Temperature: Equilibrium Properties
- 39 Instantons In Quantum Mechanics
- 40 Unstable Vacua In Quantum Field Theory
- 41 Degenerate Classical Minima And Instantons
- 42 Perturbation Series At Large Orders. Summation Methods
- 43 Multi-Instantons In Quantum Mechanics
- Index
