Trial wavefunctions that can be represented by summing over locally-coupled
degrees of freedom are called tensor network states (TNSs); they have seemed
difficult to construct for two-dimensional topological phases that possess
protected gapless edge excitations. We show it can be done for chiral states of
free fermions, using a Gaussian Grassmann integral, yielding px​±ipy​
and Chern insulator states, in the sense that the fermionic excitations live in
a topologically non-trivial bundle of the required type. We prove that any
strictly short-range quadratic parent Hamiltonian for these states is gapless;
the proof holds for a class of systems in any dimension of space. The proof
also shows, quite generally, that sets of compactly-supported Wannier-type
functions do not exist for band structures in this class. We construct further
examples of TNSs that are analogs of fractional (including non-Abelian) quantum
Hall phases; it is not known whether parent Hamiltonians for these are also
gapless.Comment: 5 pages plus 4 pages supplementary material, inc 3 figures. v2:
improved no-go theorem, additional references. v3: changed to regular article
format; 16 pages, 3 figures, no supplemental material; main change is much
extended proof of no-go theorem. v4: minor changes; as-published versio