This talk will be concerned with three new results (or a subset thereof) on the idea of grasping quantum many-body systems in terms of suitable tensor networks, such as finitely correlated states (FCS), tree tensor networks (TTN), projected entangled pair states (PEPS) or entanglement renormalization (MERA). We will first briefly introduce some basic ideas and relate the feasibility of such approaches to entanglement properties and area laws.
We will then see that (a) surprisingly, any MERA can be efficiently encoded in a PEPS, hence in a sense unifying these approaches. (b) We will also find that the ground state-manifold of any frustration-free spin-1/2 nearest neighbor Hamiltonian can be completely characterized in terms of tensor networks, how all such ground states satisfy an area law, and in which way such states serve as ansatz states for simulating almost frustration-free systems. (c) The last part will be concerned with using flow techniques to simulate interacting quantum fields with finitely correlated state approaches, and with simulating interacting fermions using efficiently contractible tensor networks.