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Table of contents
1 Introduction to non-unitary critical phenomena
1.1 Universality and CFT
1.2 The quantum Hall effect
1.3 Geometric systems and polymers
1.4 Non-unitary features
1.4.1 General considerations
1.4.2 Non-unitary representations of the Viraso algebra and negative conformal dimensions
1.4.3 Indecomposability and logarithmic correlators
1.4.4 Irrationality and non-compactness
1.4.5 PT symmetry and RG-flow
1.5 The plan of this manuscript
2 Entanglement in non-unitary critical systems
2.1 Entanglement entropy
2.1.1 Definitions
2.1.2 Conformal field theory interpretation
2.1.3 The non-unitary case: first observations
2.2 The XXZ spin chain
2.2.1 Potts model
2.2.2 Loop model formulation
2.2.3 The six-vertex model and the XXZ Hamiltonian.
2.2.4 Quantum group
2.3 Quantum group entanglement entropy
2.3.1 Pedagogical example on 2 sites
2.3.2 Entanglement in the loop model and Markov Trace
2.3.3 Definition of the quantum group entanglement entropy and motivations
2.3.4 A more complex example: 2M = 4 sites
2.3.5 Properties of the entropy
2.4 The scaling relation of the quantum group entanglement entropy
2.4.1 A brief introduction to Coulomb Gas
2.4.2 The replica trick and the modified scaling relation
2.4.3 Numerical analysis
2.5 Extensions
2.5.1 Restricted Solid-on-Solid models
2.5.2 A supersymmetric example
2.5.3 Entanglement entropy in the non-compact case
2.6 Comparisons and conclusion
2.6.1 Entanglement in non-unitary minimal models
2.6.2 The null-vector conditions in the cyclic orbifold
3 Truncations of non-compact loop models 56
3.1 The Chalker-Coddington model
3.1.1 Definition as a one-particle model
3.1.2 Supersymmetric formulation
3.1.3 The supersymmetric gl(2j2) spin chain
3.1.4 Exact results and critical exponents
3.2 The first truncation as a loop model
3.2.1 Truncations as a loop model: the case M = 1
3.2.2 An integrable deformation
3.2.3 Symmetries
3.2.4 Comparison
3.2.5 A word on the dense phase
3.2.6 Lattice observables in the network model
3.3 Higher truncations
3.3.1 The second truncation
3.3.2 Generalisation
3.3.3 Preliminary numerical results
3.4 Truncations of the Brownian motion
3.4.1 Brownian motion as a supersymmetric spin chain
3.4.2 Equivalence between oriented/unoriented lattice
3.4.3 The first truncation: self-avoiding walks
3.4.4 Hamiltonian limit
3.4.5 Symmetries in the continuum limit
3.4.6 Higher truncation of the Brownian motion
3.4.7 The multicritical point of the second truncation
4 A flow between class A and class C
4.1 Lattice model interpolating between class A and class C
4.1.1 The Spin Quantum Hall Effect as a network model
4.1.2 Second quantisation and the Hamiltonian limit
4.1.3 Choosing an interpolation
4.1.4 Loop formulation of the model
4.1.5 Percolation as a two-colours loop model
4.2 The untruncated model
4.2.1 Symmetries
4.2.2 Lyapunov exponents
4.3 Truncations
4.3.1 The phase diagram
4.3.2 Symmetries
4.3.3 The dense phase
4.3.4 Critical exponents of the critical dilute phase
5 Operators in the Potts model
5.1 Observables in the Q-state Potts model
5.1.1 Potts model and Fortuin-Kasteleyn clusters
5.1.2 Definitions and representation theory of SQ
5.1.3 Observables of one spin
5.1.4 Observables of two spins
5.1.5 Procedure for general representations
5.1.6 Internal structure and LCFT
5.2 Correlation functions
5.2.1 Symmetric observables of two spins
5.2.2 Anti-symmetric observables of two spins
5.2.3 Observables with mixed symmetry: [Q 3; 2; 1]
5.2.4 Generic case
5.3 Physical interpretation
5.3.1 Primal and secondary operators
5.3.2 Critical exponents on a cylinder
5.3.3 Numerics
5.3.4 Spin
5.4 Logarithmic correlations in 3D percolation
Conclusion
