Title: Topological Orders Having no Topological Quantum Field Theory Description

Abstract: Systems displaying quantum topological order feature robust characteristics that have been very attractive to quantum computing schemes. It has long been believed that the salient universal features of topologically ordered systems are invariably described by topological quantum field theories. In the current work, we illustrate that this is not necessarily so. Towards this end, we construct and study a rich class of two- and three-dimensional topologically ordered models featuring interacting anyons (both Abelian and non-Abelian). In these theories, the lowest excitation energies depend on the relative geometrical placement of the anyons leading to properties that cannot be described by topological quantum field theories. We examine these models by performing dualities to systems displaying conventional (i.e., Landau) orders. Our approach enables a general method for mapping general Landau type theories to topologically ordered dual models. The low-energy subspaces of our models are more resilient to thermal effects than those of surface codes.