Abstract

In these lectures, we will provide an introduction to a tensor network approach to topological defects and its applications. We will start by considering the ground states of topological fixed point models in (2+1)d, and show that their entanglement properties can be characterized by topological defects described by fusion categories. By studying the different tensor network representations of these ground states, we will obtain the lattice representation theory of these topological defects. Through a holographic reduction, we will map these (2+1)d topological models to 2d classical statistical mechanics models or 1d quantum Hamiltonians, where the lattice topological defects become symmetries of the theory. Finally, we will see how gauging these topological defect symmetries allows a precise characterization of the entanglement properties of a state in a given phase, which reduces the computational complexity of finding ground states using tensor network based methods. References: https://arxiv.org/abs/1104.5047 ;https://arxiv.org/abs/2008.11187 https://arxiv.org/abs/2408.06334