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Table of contents
1 Introduction to critical phenomena, polymer collapse and the Integer Quantum Hall Effect
1.1 Universality in macroscopic systems, and conformal eld theory
1.2 Polymer chains
1.3 The Integer Quantum Hall Eect
1.4 A step towards exact solutions : lattice models
1.5 Non-compactness
1.6 Plan of the rest of the manuscript
2 Exactly solvable models for polymers and the IQHE plateau transition
2.1 Introduction to integrability and exactly solvable models
2.1.1 Vertex models, integrability, and the Yang{Baxter equation
2.1.2 The quantum group Uq(sl2)
2.1.3 General framework
2.2 Loop models and algebraic aspects
2.2.1 Loop formulation of the six-vertex model and the Temperley{Lieb algebra
2.2.2 Representation theory of the TL algebra
2.2.3 Periodic loop and vertex models
2.2.4 Schur-Weyl duality
2.3 From polymers to the O(n) model, to the a(2) 2 model
2.4 From the IQHE to the b(1) 2 integrable model
2.4.1 The Chalker{Coddington model
2.4.2 A loop model for transport observables
2.4.3 The truncation procedure
2.4.4 Dense and dilute two-colour loop models
2.4.5 The a(2) 3 and b(1) 2 integrable chains
3 From exactly solvable models to non-compact CFTs
3.1 The algebraic Bethe Ansatz
3.1.1 Algebraic Bethe Ansatz for the six-vertex model
3.1.2 Algebraic Bethe ansatz for higher rank models
3.2 Continuum limit of the a(2) 2 model
3.2.1 Continuum limit in regimes I and II
3.2.2 First observations in regime III
3.2.3 The black hole CFT
3.2.4 The density (s) and the discrete states
3.2.5 The parafermions
3.2.6 Conclusion: the continuum limit of the a(2) 2 model
3.3 Continuum limit of the a(2) 3 model
3.3.1 Continuum limit in regimes I and II
3.3.2 Continuum limit in regime III
3.4 General solution of the a(2) n chains
3.4.1 Duality arguments for a non compact regime
3.4.2 The regimes I and II
3.4.3 The regimes III
3.5 The non compact world
3.5.1 A look at the Bethe ansatz kernels
3.5.2 The search for non compact degrees of freedom
3.5.3 Non compact theories in practice
4 Polymer collapse and non compact degrees of freedom
4.1 The VISAW phase diagram
4.2 Physical properties at the non compact point
4.2.1 Grand-canonical Monte-Carlo simulations
4.2.2 The and exponents
4.2.3 Discrete states and polymer attraction
4.3 Probing the non compact degrees of freedom
4.3.1 Relationship between the black hole and dense phase
4.3.2 Probing degrees of freedom
4.3.3 A model for K KBN
4.4 The collapse transition
5 Application to the Integer Quantum Hall Effect
5.1 Review of the results by Ikhlef et. al
5.2 Continuum limit of the b(1) n model
5.2.1 Bethe ansatz solution in regime IV
5.2.2 b(1) 2 , a(2) 3 , and Pohlmeyer reduction
5.2.3 Critical exponents of the truncated model
5.3 Continuous exponents at the IQHE transitions
5.3.1 GL(2j2) interpretation of point-contact conductances
5.3.2 Loop interpretation
5.3.3 Pure scaling observables
Conclusion
Bibliography
