Linnea Grans-Samuelsson

Théories conformes non unitaires et problèmes géométriques : approche par le réseau: Dans cette thèse, nous étudions les théories conformes des champs non unitaires à deux dimensions qui apparaissent dans la limite continue de certains modèles critiques sur réseau, y compris les modèles de Potts et O(n). Ces modèles ont des applications à des problèmes géométriques tels que la percolation et les marches auto-évitantes (polymères). Pour traiter les difficultés techniques causées par la non-unitarité, nous considérons une approche par le réseau, qui inclut des techniques telles que l'ansatz de Bethe, la théorie des représentations de l'algèbre affine de Temperley-Lieb et l'utilisation d'une discrétisation de l'algèbre de Virasoro (les générateurs de Koo-Saleur). Nous fournissons une formulation graphique des fonctions de corrélation dans les modèles minimaux RSOS et les comparons à des quantités similaires dans le modèle de Potts. Cela nous permet d'expliquer pourquoi une conjecture antérieure pour les fonctions à quatre points de Potts, bien qu'incorrecte, ait donné des résultats en bon accord avec les simulations numériques. Nous détaillons les connexions entre les modèles RSOS et les systèmes anyoniques, en montrant que le calcul des fonctions de corrélation se réduit à l'évaluation de certains diagrammes de fusion anyoniques. La partie principale de la thèse est consacrée à l'étude des modules de Virasoro présents dans le modèle à six vertex et dans les modèles de Potts et de boucles à charge centrale générique (irrationnelle). Ceci est fait principalement en utilisant les générateurs de Koo-Saleur. Nous trouvons dans le modèle à six vertex à la fois des modules de Verma et des modules co-Verma, tandis que dans les modèles de Potts et de boucles, la principale découverte est la présence de modules logarithmiques correspondants à des cellules de Jordan de rang deux. Ces cellules de Jordan ne sont pas présentes à taille finie, mais apparaissent seulement dans la limite continue. Nous montrons également la convergence des générateurs de Koo-Saleur vers les générateurs de Virasoro dans une procédure à double limite appelée limite d'échelle, où l'ordre des limites est crucial. Enfin, l'information sur les modules logarithmiques est utilisée dans le bootstrap du modèle O(n) pour des valeurs complexes génériques de n, en utilisant des blocs conformes logarithmiques dans les équations de croisement. Nous calculons numériquement les fonctions à quatre points impliquant les opérateurs les plus simples et constatons que le nombre de solutions aux équations de croisement est compatible avec la théorie des représentations de O(n).

Topological defect lines (TDLs) in two-dimensional conformal field theories (CFTs) are standard examples of generalized symmetries in quantum field theory. Integrable lattice incarnations of these TDLs, such as those provided by spin/anyonic chains, provide a crucial playground to investigate their properties, both analytically and numerically. Here, a family of parameter-dependent integrable lattice models is presented, which realize different TDLs in a given CFT as the parameter is varied. These models are based on the general quantum-inverse scattering construction, and involve inhomogeneities of the spectral parameter. Both defect hamiltonians and (defect) line operators are obtained in closed form. By varying the inhomogeneities, renormalization group flows between different TDLs (such as the Verlinde lines associated with the Virasoro primaries (1, s) and (r, 1) in diagonal minimal CFTs) are then studied using different aspects of the Bethe-ansatz as well as ab-initio numerical techniques. Relationships with the anisotropic Kondo model as well as its non-Hermitian version are briefly discussed.

Topological/perfectly-transmissive defects play a fundamental role in the analysis of the symmetries of two dimensional conformal field theories (CFTs). In the present work, spin chain regularizations for these defects are proposed and analyzed in the case of the three-state Potts CFT. In particular, lattice versions for all the primitive defects are presented, with the remaining defects obtained from the fusion of the primitive ones. The defects are obtained by introducing modified interactions around two given sites of an otherwise homogeneous spin chain with periodic boundary condition. The various primitive defects are topological on the lattice except for one, which is topological only in the scaling limit. The lattice models are analyzed using a combination of exact diagonalization and density matrix renormalization group techniques. Low-lying energy spectra for different defect Hamiltonians as well as entanglement entropy of blocks located symmetrically around the defects are computed. The latter provides a convenient way to compute the g-function which characterizes various defects. Finally, the eigenvalues of the line operators in the “crossed channel” and fusion of different defect lines are also analyzed. The results are all in agreement with expectations from conformal field theory.

Topological/perfectly-transmissive defects play a fundamental role in the analysis of the symmetries of two dimensional conformal field theories (CFTs). In the present work, spin chain regularizations for these defects are proposed and analyzed in the case of the three-state Potts CFT. In particular, lattice versions for all the primitive defects are presented, with the remaining defects obtained from the fusion of the primitive ones. The defects are obtained by introducing modified interactions around two given sites of an otherwise homogeneous spin chain with periodic boundary condition. The various primitive defects are topological on the lattice except for one, which is topological only in the scaling limit. The lattice models are analyzed using a combination of exact diagonalization and density matrix renormalization group techniques. Low-lying energy spectra for different defect Hamiltonians as well as entanglement entropy of blocks located symmetrically around the defects are computed. The latter provides a convenient way to compute the $g$-function which characterizes various defects. Finally, the eigenvalues of the line operators in the "crossed channel'' and fusion of different defect lines are also analyzed. The results are all in agreement with expectations from conformal field theory.

In two-dimensional statistical physics, correlation functions of the O(N) O ( N ) and Potts models may be written as sums over configurations of non-intersecting loops. We define sums associated to a large class of combinatorial maps (also known as ribbon graphs). We allow disconnected maps, but not maps that include monogons. Given a map with n n vertices, we obtain a function of the moduli of the corresponding punctured Riemann surface. Due to the map’s combinatorial (rather than topological) nature, that function is single-valued, and we call it an n n -point correlation function. We conjecture that in the critical limit, such functions form a basis of solutions of certain conformal bootstrap equations. They include all correlation functions of the O(N) O ( N ) and Potts models, and correlation functions that do not belong to any known model. We test the conjecture by counting solutions of crossing symmetry for four-point functions on the sphere.

In two-dimensional statistical physics, correlation functions of the O(N) and Potts models may be written as sums over configurations of non-intersecting loops. We define sums associated to a large class of combinatorial maps (also known as ribbon graphs). We allow disconnected maps, but not maps that include monogons. Given a map with n vertices, we obtain a function of the moduli of the corresponding punctured Riemann surface. Due to the map's combinatorial (rather than topological) nature, that function is single-valued, and we call it an n-point correlation function. We conjecture that in the critical limit, such functions form a basis of solutions of certain conformal bootstrap equations. They include all correlation functions of the O(N) and Potts models, and correlation functions that do not belong to any known model. We test the conjecture by counting solutions of crossing symmetry for four-point functions on the sphere.