Symmetries and field tensor network states

We study the interplay between symmetry representations of the physical and virtual space on the class of tensor network states for critical spins systems known as field tensor network states (fTNSs). These are by construction infinite-dimensional tensor networks whose virtual space is described by a conformal field theory (CFT). We can represent a symmetry on the physical index as a commutator with the corresponding CFT current on the virtual space. By then studying this virtual space representation we can learn about the critical symmetry-protected topological properties of the state, akin to the classification of symmetry-protected topological order for matrix product states. We use this to analytically derive the critical symmetry-protected topological properties of the two ground states of the Majumdar-Ghosh point with respect to the previously defined symmetries.