A practical introduction to tensor networks: Matrix product states and projected entangled pair states

This is a partly non-technical introduction to selected topics on tensor network methods, based on several lectures and introductory seminars given on the subject. It should be a good place for newcomers to get familiarized with some of the key ideas in the field, specially regarding the numerics. After a very general introduction we motivate the concept of tensor network and provide several examples. We then move on to explain some basics about Matrix Product States (MPS) and Projected Entangled Pair States (PEPS). Selected details on some of the associated numerical methods for 1d and 2d quantum lattice systems are also discussed. •A practical introduction to selected aspects of tensor network methods is presented.•We provide analytical examples of MPS and 2d PEPS.•We provide basic aspects on several numerical methods for MPS and 2d PEPS.•We discuss a number of applications of tensor network methods from a broad perspective.