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Trial wavefunctions that can be represented by summing over locally-coupleddegrees of freedom are called tensor network states (TNSs); they have seemeddifficult to construct for two-dimensional topological phases that possessprotected gapless edge excitations. We show it can be done for chiral states offree fermions, using a Gaussian Grassmann integral, yielding $p_x \pm i p_y$and Chern insulator states, in the sense that the fermionic excitations live ina topologically non-trivial bundle of the required type. We prove that anystrictly short-range quadratic parent Hamiltonian for these states is gapless;the proof holds for a class of systems in any dimension of space. The proofalso shows, quite generally, that sets of compactly-supported Wannier-typefunctions do not exist for band structures in this class. We construct furtherexamples of TNSs that are analogs of fractional (including non-Abelian) quantumHall phases; it is not known whether parent Hamiltonians for these are alsogapless.

Tensor network trial states for chiral topological phases in two dimensions and a no-go theorem in any dimension