Indecomposability in field theory and applications to disordered systems and geometrical problems
Fri, Sep. 27th 2013, 14:00-18:00
Salle Claude Itzykson, Bât. 774, Orme des Merisiers
Logarithmic Conformal Field Theories (LCFTs) are crucial for describing the critical behavior of a variety of physical systems. These include phase transitions in disordered non-interacting electronic systems (such as the transition between plateaus in the integer quantum Hall effect), disordered critical points in classical statistical models (such as the random bond Ising model), or critical geometrical models (such as polymers and percolation). LCFTs appear when one has to give up the unitarity condition, which is natural in particle physics applications, but not in statistical mechanics and condensed matter physics. Without unitarity, the powerful algebraic approach of conformal invariance encounters formidable technical difficulties due to ``indecomposability''. This in turn yields logarithmic corrections to the power-law correlations at the critical point, and prevents the use of general classification techniques that have proven so powerful in the unitary case. par The goal of this thesis is to understand LCFTs by studying their lattice regularizations, the crucial point being that most algebraic complications due to indecomposability occur in finite size systems as well. Our approach is thus to consider critical statistical models with non-diagonalizable transfer matrices, or gapless quantum spin chains with non-diagonalizable hamiltonians, and to study their scaling limit by utilizing a variety of algebraic, numerical and integrable techniques. We show how to measure numerically universal parameters that characterize the indecomposable representations of the Virasoro algebra which emerge in the thermodynamic limit. An extensive understanding of a wide class of lattice models allows us to conjecture a tentative classification of all possible (chiral) LCFTs with Virasoro symmetry only. This approach is partially extended to the bulk case, for which we discuss how the longstanding bulk CFT formulation of percolation can be tackled along these lines. We also argue that boundary and periodic lattice models can be related algebraically only in the case of minimal models, and we work out the consequences for the underlying boundary and bulk field theories. Several concrete applications to disordered systems and geometrical problems are discussed, and we uncover a large class of geometrical observables in the Potts model that behave logarithmically at the critical point. \ \ Supervisors: Jesper Jacobsen and Hubert Saleur. \ PhD committee: Denis Bernard, John Cardy, Matthias Gaberdiel, Jesper Jacobsen, Nicholas Read, Hubert Saleur and Jean-Bernard Zuber.