Renormalization group flow and fixed-point in tensor network representations

Abstract

In this thesis, we present a novel method combining energy-based finite-size scaling with tensor network renormalization (TNR) to study phase transitions in lattice models. This approach effectively calculates running coupling constants and reduces the numerical errors typically associated with TNR, thus requiring fewer renormalization group (RG) steps and less computational resources. Our methodology, contrasting with traditional methods, doesn't depend on large system sizes, making it efficient and robust against simulation scale challenges. We also explore the origins of numerical errors in TNR from a field-theoretical perspective, focusing on how these errors scale with the approximation parameter $D$. This understanding is crucial for error management in simulations. Moreover, we investigate the tensor structure of fixed points in lattice models, addressing challenges from finite bond dimensions using an analytical approach involving conformal mappings. We discover that the tensor elements of fixed-point tensors align with four-point functions of primary operators in conformal field theory (CFT), demonstrating a significant link between CFT and lattice models. This finding underscores the universality of non-trivial infrared physics at the lattice level, bridging theoretical concepts with practical computations in lattice models, and offering deeper insights into the universal aspects of critical phenomena.