The space of two dimensional quantum field theories: deformations of conformal models

In the set of conformal models in two dimensions there are some special cases, known as minimal models, which admit a full control of the Hilbert space of the theory. Starting from these models, we will study a numerical method (the Truncated conformal space approach) to explore, non-perturbatively, the space of deformations of the simpler minimal models. From the point of view of the renormalization group these deformations span the space of two dimensional quantum field theories. We will test the method on integrable deformations comparing exact results coming from integrability with our numerical analysis. Finally, we will focus on the purely imaginary magnetic deformation of the tricritical Ising model: the Yang-Lee formulation of statistical mechanics suggests the presence of a (non-unitary) conformal field theory along this RG flow. In the simpler case, i.e. the Ising model, this conformal theory is believed to be the Yang-Lee conformal model as proposed by J. Cardy; indeed recent numerical studies by Zamolodchikov and collaborators seem to confirm this ansaz. The tricritical case is less studied and the theory is not identified.