  Despite the success in describing a range of quantum many-body states using
tensor networks, there is a no-go theorem that rules out strictly local tensor
networks as topologically nontrivial groundstates of gapped parent Hamiltonians
with short-range (including exponentially decaying) couplings. In this work, we
show that for free fermions, strictly local tensor networks may describe
nonzero temperature averages with respect to gapped Hamiltonians with
exponentially decaying couplings. Parent Hamiltonians in this sense may be
constructed for any dimensionality and without any obstructions due to their
topology. Conversely, we also show that thermal averages with respect to
gapped, strictly short-range free-fermion Hamiltonians can be calculated by
tensor networks whose links decay exponentially with distance. We also describe
a truncation-reconstruction scheme for such tensor networks that leads to a
controlled approximation of exact averages in terms of a sequence of related
thermal averages. We illustrate our scheme on the two-dimensional Haldane
honeycomb model considering both topological and nontopological phases.
