LAUSR

We discuss in detail algorithms for implementing tensor network renormalization (TNR) for the study of classical statistical and quantum many-body systems. First, we recall established techniques for how the partition function of a $2D$ classical many-body system or the Euclidean path integral of a $1D$ quantum system can be represented as a network of tensors, before describing how TNR can be implemented to efficiently contract the network via a sequence of coarse-graining transformations. The efficacy of the TNR approach is then benchmarked for the $2D$ classical statistical and $1D$ quantum Ising models; in particular the ability of TNR to maintain a high level of accuracy over sustained coarse-graining transformations, even at a critical point, is demonstrated.