Ungappable edge theories with finite dimensional Hilbert spaces

TL;DR

This paper constructs finite-dimensional edge theories for certain fermionic Abelian topological phases, revealing some boundaries that cannot be gapped, with implications for understanding topological edge states.

Contribution

It introduces a novel class of edge theories with finite-dimensional Hilbert spaces for Abelian topological phases, including ungappable boundaries, derived via impurity scattering models.

Findings

Edge theories have finite Hilbert spaces for finite systems.

Some boundaries are inherently ungappable by local interactions.

Ground state degeneracy scales exponentially with impurity number.

Abstract

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, $\kappa_H = 0$.