Boundary conditions in some non-unitary conformal field theories
Tue, Sep. 07th 2010, 15:00
Amphi Claude Bloch, Bât. 774, Orme des Merisiers
Understanding the surface effects, as well as finite size effects, is crucial in the study of critical phenomena. Indeed, whereas theoretical models are most often built in very large systems with periodic boundary conditions, the samples which are used in experimental physics are finite and have some boundaries. At a critical point, the correlation length diverges, so it can be comparable to, or even larger than the size of the sample. Thus, surface effects are important even in the bulk, and they cannot be negleted in the sample. Our understanding of critical phenomena at a surface has made the same progress as its bulk counterpart. In particular, in two dimensions, conformally invariant field theories are now extremely powerful tools which are used to describe phase transitions in a non-perturbative way. In this context, the study of surface critical phenomena has produced numerous new exact results, such as boundary critical exponents or correlation functions in several critical models. \par In this thesis, we are interested in some statistical field theories in two dimensions, with non-local degrees of freedom. For example, polymers in a good solvant are described by such theories. One can try to turn these theories into local ones, but the price to pay is that we get negative or even complex Boltzmann weights: these theories are then non-unitary. We are interested in surface effects, and in finding out which boundary conditions are compatible with conformal invariance in these theories. Our strategy does not involve an axiomatic approach, but rather relies on concrete lattice models which have an interesting scaling limit. In these models, the configurations of the model are given in terms of geometrical objects that are non-local. Our results can be reformulated in the context of the (stochastic) Schramm-Loewner Evolution (SLE), which gives a description of these objects directly in the scaling limit. \par In some cases, the models we study can be considered as toy-models for logarithmic theories.
Indeed, it should be easier to understand the algebraic structure which underlies a particular theory in the boundary case, because in the bulk theory the actions of the chiral and anti-chiral algebras are mixed.
For example, it must be possible to extract universal parameters which caracterize the indecomposability of representations of the Virasoro algebra. Some other results in this thesis can be used to compute some universal quantities which appear at quantum critical points of 1+1D systems (e.g. gapless spin chains), such as entanglement entropies or fidelities, in particular in finite-size systems.