Some steps toward exact critical RG fixed point

Is there an exact (truncation-free) tensor RG map that has a fixed point corresponding to the 2D critical Ising model and is well-defined in its neighborhood? In this talk, I will review two recently developed techniques that make this question approachable: rotation-modified RG maps and hat-tensors. First, I will explain how incorporating rotation into the RG map enables the use of the Newton method. This allows for a more efficient numerical search for tensor RG fixed points and paves the way for applying the Earle-Hamilton theorem to prove the existence of fixed points. Then, I will present hat-tensors and hat-RG maps, which allow us to track finite-dimensional “bounding boxes” for infinite-dimensional tensors. The talk is based on [2408.10312] and [2506.03247].