How many kinds of topological degeneracy are there?

Question

Here I want to summarize the various kinds of topological ground-state degeneracy in condensed matter physics and want to know whether there exists any other kind of topological degeneracy. For convenience, let's consider a 2D lattice system with $N$ lattice sites, and we may consider the finite system with open boundary condition(OBC) or periodic boundary condition(PBC). Now there are two kinds of topological degeneracy:

(1)The topological degeneracy is approximate for finite $N$ no matter with OBC or PBC, and it becomes exact degenerate only under the thermodynamic limit($N\rightarrow \infty $). Example: FQHE.

(2)The topological degeneracy(>1) is exact for any finite $N$ with PBC(on a torus), and it's nondegenerate for any finite $N$ with OBC. Example: Kitaev's toric code model.

Is there any other kind(in the above sense) of topological degeneracy?

Answer 1

Topological degeneracy is only defined in the thermodynamic limit on a closed manifold. The ground state degeneracy of a finite-sized system or on an open manifold is not "topological", and can not be called topological degeneracy.

Considering your examples. (1) The ground state degeneracy is ill-defined with open boundary condition. Because there might be gapless edge modes on the boundary (which is indeed the case for FQHE), such that low-energy states form a continuum and the "ground state" can not be separate out. (2) The exact degeneracy for toric code model on a finite-sized system is not robust against local perturbation, and therefore not topological. Adding an $h \sigma_x$ term to the toric code Hamiltonian is enough to lift the degeneracy for finite-sized system. It is only because that the toric code model is fine-tuned to an ideal point that the degeneracy happens to be exact. In reality, finite-sized system is not topologically robust, so it makes no sense to classify the degeneracy for a finite-sized system.