Applying FSS in absence of a small coupling

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Table of contents

Contents
1 Introduction
2 Introduction to the functional renormalization framework
2.1 The role of correlations
2.1.1 Mean eld theory applied to gas-liquid and uni-axial ferromagnetic systems
2.1.2 Landau theory
2.1.3 Range of application of Landau theory
2.1.4 Going beyond mean eld theory
2.2 Perturbative Renormalization without eld theory: a rst conceptual step towards functional renormalization
2.2.1 A one loop calculation
2.2.2 Eective scale-dependent parameters
2.2.3 A divergent product
2.2.4 An exact solution from a rst order correction using the renormalization group
2.2.5 An improved approximation using the renormalization group :
2.2.6 Charge beta function
2.2.7 A comparison of the renormalization group and the variational approach on an approximation of a non trivial second order dierential equation
2.3 Non perturbative Renormalization
2.3.1 Exact RG equations
2.3.2 The 􀀀 ow as an interpolation function
2.3.3 The 􀀀 ow as an RG improved one loop calculation
2.3.4 Diusive nature of the RG ow
2.3.5 Phase transition and stability analysis
2.4 Approximation schemes
2.4.1 FSS as a convergence accelerator
2.4.2 Applying FSS in absence of a small coupling
2.4.3 NPRG approximation schemes
3 Application of the functional renormalisation group to models
3.1 O(N) models and the Bardeen-Moshe-Bander phenomenon
3.1.1 O(N) models
3.1.2 Multi-critical points of the O(N) model
3.1.2.1 Multi-critical points within the framework of Landau theory
3.1.2.2 Multicritical xed-points in the O (N) model
3.1.3 Bardeen-Moshe-Bander phenomenon using standard eld theory techniques
3.1.3.1 Large N analysis: leading order
3.1.3.2 Large N analysis: order 1=N
3.1.4 BMB phenomenon at the level of the LPA
3.1.5 Improving the LPA result
3.1.6 Generalization to all upper multicritical dimensions
3.1.7 Exact order 1/N equations
3.1.8 BMB phenomenon at order 2 of the derivative expansion
3.1.9 Physical interpretation of cusped xed-points
3.1.10 Extension of the BMB phenomenon to moderate N and non trivial homotopies in (N; d) space
4 Conclusion
A Van der Waals Phase diagram
B Discussion on Euler product
C Counter terms
D Formal derivation of the 􀀀 ow
E FSS RG via rescalings
F Derivative expansion without an underlying eective action
G Fluctuation dissipation relations
H Multicritical phase diagram
I 1/N expansion for the tricritical function
J LPA Polchinski and 􀀀 ow equation
K Derivation of () at LPA
L LPA singular solutions as weak solutions
M Singular perturbation theory for the LPA
N Boundary layer analysis of xed-point SG
O LPA equivalence of Polchinski and 􀀀 ow for Litim regulator
P SG eigenvalues
Q Coupling to (d;N) space mapping for all multicritical dimensions
R Large N ow equations at order 2 of the derivative expansion
Bibliography