We will briefly review Wilson-Kadanoff type renormalization group (RG) maps for Ising spin systems and the lack of progress in proving that there is a non-trivial fixed point for these maps. The Ising model can be written as a tensor network, and RG maps can be defined in the tensor network formalism. Numerical studies of these tensor network RG maps by many groups have been remarkably successful in two dimensions. In joint work with Slava Rychkov we have begun a rigorous study of such tensor network RG maps. In particular we proved that in two dimensions there is a high temperature fixed point tensor which is locally stable. We have also proved results in the low temperature phase. Here there are two stable fixed points and one unstable fixed point which is related to behavior near the phase coexistence curve. Our long range goal is to prove the existence of a non-trivial fixed point which describes the second order phase transition in the Ising model.
Sylvain Carrozza, Luca Lionni, Fabien Vignes-Tourneret