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

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 $p_x \pm i p_y$ 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

Edge excitations of paired fractional quantum Hall states

The Hilbert spaces of the edge excitations of several ``paired'' fractional quantum Hall states, namely the Pfaffian, Haldane-Rezayi and 331 states, are constructed and the states at each angular momentum level are enumerated. The method is based on finding all the zero energy states for those Hamiltonians for which each of these known ground states is the exact, unique, zero-energy eigenstate of lowest angular momentum in the disk geometry. For each state, we find that, in addition to the usual bosonic charge-fluctuation excitations, there are fermionic edge excitations. The edge states can be built out of quantum fields that describe the fermions, in addition to the usual scalar bosons (or Luttinger liquids) that describe the charge fluctuations. The fermionic fields in the Pfaffian and 331 cases are a non-interacting Majorana (i.e., real Dirac) and Dirac field, respectively. For the Haldane-Rezayi state, the field is an anticommuting scalar. For this system we exhibit a chiral Lagrangian that has manifest SU(2) symmetry but breaks Lorentz invariance because of the breakdown of the spin statistics connection implied by the scalar nature of the field and the positive definite norm on the Hilbert space. Finally we consider systems on a cylinder where the fluid has two edges and construct the sectors of zero energy states, discuss the projection rules for combining states at the two edges, and calculate the partition function for each edge excitation system at finite temperature in the thermodynamic limit. It is pointed out that the conformal field theories for the edge states are examples of orbifold constructions.Comment: 44 pages, requires RevTeX, no figure