Ungappable edge theories with finite-dimensional Hilbert spaces

We construct a class of edge theories for a family of fermionic Abelian topological phases with K matrices of the form K=k100-k2, where k1,k2>0 are odd integers. Our edge theories are notable for two reasons: (i) they have finite-dimensional Hilbert spaces (for finite-sized systems) and (ii), depending on the values of k1,k2, some of the edge theories describe boundaries that cannot be gapped by any local interaction. The simplest example of such an ungappable boundary occurs for (k1,k2)=(1,3), which is realized by the ν=2/3 fractional quantum Hall state. We derive our edge theories by starting with the standard chiral boson edge theory, consisting of two counterpropagating chiral boson modes, and then introducing an array of pointlike impurity scatterers. We solve this impurity model exactly in the limit of infinitely strong impurity scattering, and we show that the energy spectrum consists of a gapped phonon spectrum together with a ground-state degeneracy that scales exponentially with the number of impurities. This ground-state subspace forms the Hilbert space for our edge theory. We believe that similar edge theories can be constructed for any Abelian topological phase with vanishing thermal Hall coefficient, κH=0.