NPRG Approach to Polymerized Membranes

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Table of contents

Abstract
Acknowledgements
Chapter 1 Introduction
Chapter 2 Non-Perturbative Renormalization Group
2.1 A Brief Historical Introduction
2.2 Wilson Renormalization Group
2.2.1 Kadanoff’s Block Spin
2.2.2 Wilson Momentum Shell Integration
2.2.3 Polchinski and Proof of Renormalizability
2.3 Wetterich Renormalization Group
2.3.1 Effective Average Action
2.3.2 The Wetterich Equation
2.3.3 Approximations
2.3.3.1 Derivative Expansion
2.3.3.2 Field Expansion
2.3.3.3 Combination of the Derivative Expansion and the Field Expansion
2.3.3.4 Blaizot-Mendez-Wschebor Approximation
2.3.4 Optimisation and Cut-off Function Choice
2.3.5 The O(n)-model
2.3.5.1 Propagator
2.3.5.2 Definitions of the Coupling Constants
2.3.5.3 Derivation of the Flow Equations
2.3.5.4 Weak-coupling Expansion
2.3.5.5 Low-Temperature Expansion
2.3.5.6 Large-n Expansion
2.3.6 Conclusion
Appendix Chapter A Threshold Functions
Chapter 3 Membranes
3.1 Introduction
3.2 Differential Geometry of Membranes
3.2.1 Basic Definitions and Some Fundamental Properties
3.2.2 Monge Parametrization
3.3 Deformations
3.4 Long-Range Behaviour of Fluid Membranes
3.4.1 The Model
3.4.2 Fluid Membranes in Monge Parametrization
3.5 Polymerized Membranes
3.5.1 The Model
3.5.2 Mean Field Theory
3.5.3 Perturbative RG for the Crumpled-to-Flat Transition
3.5.4 The normal-normal Correlation Function in the Harmonic Approximation
3.5.5 Self-Consistent Screening Approximation (SCSA)
3.5.6 Conclusion
3.6 NPRG Approach to Polymerized Membranes
3.6.1 The propagator in Fourier Space
3.6.2 The Minimum Configuration
3.6.3 The Flow Equations of u and v and the Anomalous Dimension k
3.6.4 Derivation of the Flow Equations
3.6.4.1 The Effective Action
3.6.4.2 The Configuration
3.6.4.3 Propagator at the minimum
3.6.4.4 Flow Equations of u and v
3.6.4.5 Flow of 2
3.6.4.6 Definitions of the Coupling Constants
3.6.5 Crumpled to Flat Transition
3.6.6 Symmetry Breaking, Goldstone Bosons and Flat Phase
3.7 Conclusion
Appendix Chapter B Cayley-Hamilton Theorem and Faddeev-Leverrier Algorithm
Appendix Chapter C Derivatives of the Flow of Ueff
C.1 Derivative of the Propagator
Appendix Chapter D New Definition of the Minimum Configuration
Appendix Chapter E Threshold Functions
Chapter 4 Anisotropic Membranes
4.1 Introduction
4.2 Anisotropic Scaling Behaviour
4.3 Perturbative RG
4.4 Non-Perturbative Approach
4.4.1 The propagator
4.4.2 Flow equation of y
4.4.3 Flow equations
4.5 Physical Results
4.6 Conclusion
Appendix Chapter F Threshold Functions
Chapter 5 Lifshitz Critical Behaviour
5.1 Introduction
5.2 The Model
5.3 Anisotropic Scale Invariance
5.4 Critical Dimensions
5.5 Perturbative RG
5.5.1 Weak-Coupling -Expansion
5.5.2 Large-n Expansion
5.6 NPRG Approach
5.6.1 Lowest Order of the Derivative Expansion
5.6.2 Flow Equations
5.6.3 Upper Critical Dimension duc = 4 + m
5.6.4 Lower Critical Dimension dlc = 2 + m
5.7 Higher Order Expansion and Physical Results
5.8 Conclusion
Appendix Chapter G Threshold Functions
Chapter 6 Disordered Membranes
6.1 Introduction
6.2 Replica Formalism
6.3 The Model
6.4 Perturbative RG
6.5 NPRG
6.5.1 Effective Action
6.5.2 Propagator
6.5.3 Flow Equations
6.5.4 Conclusion
Conclusion