What are the examples of a one dimensional spin chain, with local interaction and degenerate ground space (degeneracy may be a function of n, such as log(n) etc, where n is the length of chain) and a gap to the higher energy states, where gap is independent of n?
Edit: I managed to find an example with ground state degeneracy scaling as $N+1$, where $N$ is length of chain. Hamiltonian is projector onto the state $ | 01\rangle - q|10\rangle$. Gap depends on parameter q and is gapped for q<1 . Reference: arxiv.org/abs/cond-mat/9512120