In this course we will introduce the main families of tensor networks used to describe gapped systems and their symmetries in one and two-dimensional lattices.

We will focus on the analytical questions that can be solved using the framework of tensor networks. This framework offers a powerful approach to encode global properties at the local level of their constituents: the local tensors. This local encoding has been key to the classification of gapped quantum phases in one-dimensional systems, both with and without symmetries, as well as in two-dimensional systems exhibiting topological order.

We will begin by defining matrix product states (the main family in 1D systems), the construction of their parent Hamiltonians and deriving a proof of their fundamental theorem. From this starting point, we will show how the classification of phases for group symmetries is encoded at the local level. We will then move beyond group symmetries to the so-called non-invertible symmetries (described by fusion categories), which are represented by matrix product operators (MPOs), and extend the phase classification to this setting.

We will also cover 2D systems by using the framework of projected entangled pair states (PEPS), exploring how they encode topological order and its interplay with global symmetries. We will highlight recent developments of the fundamental theorem applied to topologically ordered PEPS.

Time permitting, we will also discuss how gauging operators are expressed in the language of tensor networks, how they can be used to construct holographic maps and we will generalize the favorable properties of the parent Hamiltonian for W-like states. The lectures will be accompanied by simple examples and will indicate the open questions in the field.