  We construct a new class of edge theories for a family of fermionic Abelian
topological phases with $K$-matrices of the form $K = \begin{pmatrix} k_1 & 0
\\ 0 & - k_2 \end{pmatrix}$, where $k_1, k_2 > 0$ 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 $k_1,
k_2$, 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 $(k_1, k_2) = (1, 3)$, which is realized by the $\nu = 2/3$ FQH
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 infinite 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, $\kappa_H = 0$.
