Real-Space Renormalization Process using High-Order Tensor Renormalization Group for CFT properties*

In the field of tensor networks, the idea of the real-space renormalization group (RG) is used to compute various properties for lattice models at the thermodynamic limit. High Order Tensor Renormalization Group (HOTRG) is such a method that iteratively combines every two lattice sites into one. Short-distance entanglement removal methods such as Graph Independent Local Truncation (GILT) are crucial for those methods to succeed when the system is at a critical point. In this work, we study the numerical results of HOTRG to see if it can be conceptually understood as a real-space renormalization process. Firstly, we confirmed that the fixed point tensor of HOTRG contains information about the corresponding Conformal Field Theory (CFT), which describes the behavior of a critical system. From the fixed point tensor, we can extract CFT data, including conformal dimensions and operator product coefficients. Secondly, we study the behavior of defects under HOTRG. In particular, we found that the point-like defects are smeared if proper short-distance entanglement removal is applied. This is expected since the concept of RG is to remove the short-distance behavior. We also discussed whether HOTRG can effectively compute the N-point function. Our results provide a better understanding of the capacity and limitations of the tenor renormalization group scheme in coarse-graining defect tensors and throw light on a better understanding of the role of entanglement on critical quantum behaviors.