A hierarchy of topological tensor network states

We present a hierarchy of quantum many-body states among which many examples of topological order can be identified by construction. We define these states in terms of a general, basis-independent framework of tensor networks based on the algebraic setting of finite-dimensional Hopf C*-algebras. At the top of the hierarchy we identify ground states of new topological lattice models extending Kitaev's quantum double models [Ann. Phys. 303, 2 (2003)10.1016/S0003-4916(02)00018-0]. For these states we exhibit the mechanism responsible for their non-zero topological entanglement entropy by constructing an entanglement renormalization flow. Furthermore, we argue that the hierarchy states are related to each other by the condensation of topological charges.