  Systems displaying quantum topological order feature robust characteristics
that are very attractive to quantum computing schemes. Topological quantum
field theories have proven to be powerful in capturing the quintessential
attributes of systems displaying topological order including, in particular,
their anyon excitations. Here, we investigate systems that lie outside this
common purview, and present a rich class of models exhibiting topological
orders with distance-dependent interacting anyons. As we illustrate, in some
instances, the gapped lowest-energy excitations are comprised of anyons that
densely cover the entire system. This leads to behaviors not typically
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 generic Landau-type theories
to dual models with topological order of the same spatial dimension. The
low-energy subspaces of our models can be made more resilient to thermal
effects than those of surface codes.
