  Recently, a class of tensor networks called isometric tensor network states
(isoTNS) was proposed which generalizes the canonical form of matrix product
states to tensor networks in higher dimensions. While this ansatz allows for
efficient numerical computations, it remained unclear which phases admit an
isoTNS representation. In this work, we show that two-dimensional string-net
liquids, which represent a wide variety of topological phases including
discrete gauge theories, admit an exact isoTNS representation. We further show
that the isometric form can be preserved after applying a finite depth local
quantum circuit. Taken together, these results show that long-range
entanglement by itself is not an obstruction to isoTNS representation and
suggest that all two-dimensional gapped phases with gappable edges admit an
isoTNS representation.
