  We demonstrate that perturbative expansions for quantum many-body systems can
be rephrased in terms of tensor networks, thereby providing a natural framework
for interpolating perturbative expansions across a quantum phase transition.
This approach leads to classes of tensor-network states parametrized by few
parameters with a clear physical meaning, while still providing excellent
variational energies. We also demonstrate how to construct perturbative
expansions of the entanglement Hamiltonian, whose eigenvalues form the
entanglement spectrum, and how the tensor-network approach gives rise to order
parameters for topological phase transitions.
