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Non-compact conformal field theory and lattice models - the open case

Abstract : It is well known that lattice systems undergoing second-order phase transitions are described by Conformal Field Theories (CFTs) in the continuum limit. This thesis revolves around the study of open critical lattice models and their descriptions in the continuum limit by boundary CFTs, and is particularly concerned with models whose CFT descriptions have certain exotic properties such as being non-compact, a property identified by the appearance of a continuum of critical exponents in the language of statistical mechanics, and by the appearance of a continuum of conformal dimensions in the language of CFT. Tools from integrability such as the Bethe Ansatz, as well as numerical techniques such as exact diagonalisation are used to move between the lattice and field theory descriptions of the models under consideration.Particular focus is applied to the Potts model at its antiferromagnetic critical point, so-called due to the sign of the coupling constant at this critical point. A starting point in the analysis presented here is that new conformal boundary conditions in the antiferromagnetic Potts model are found and are shown to result in the appearance of the discrete character of the Euclidean Black Hole CFT on the lattice. Further study involving the lattice algebra representation theory results in an identity relating this discrete character to the string functions from the parafermion CFT.Motivated by the potential to apply the tools of integrability, the antiferromagnetic Potts model - and in particular its description as a staggered vertex model - is shown to map exactly to an integrable model constructed from the so-called D_2^2 algebra. This paves the way for an exact solution of the antiferromagnetic Potts model with two independent conformally invariant boundary conditions, both of which have convenient interpretations when the problem is formulated as a loop model. The continuum limit of the model with one of these boundary conditions is found to be non-compact, and a boundary renormalisation group flow is observed from a non-compact boundary CFT to a compact one.
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Niall Robertson. Non-compact conformal field theory and lattice models - the open case. Mathematical Physics [math-ph]. Université Paris-Saclay, 2020. English. ⟨NNT : 2020UPASS099⟩. ⟨tel-02909756⟩

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