# Example of gapped spin chain with degenerate ground space

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

## 2 Answers

In the XXZ spin chain, there exists some regime where the ground state becomes doubly degenerated in the infinite volume limit. This degeneracy produces a gap in energy between the ground state and the first excited state. As far as I know, on the finite chain, there is no degeneracy at all : the gap is therefore not independent of $n$. I don't know if such system exists.

The classical Ising model $$H=\sum_i \sigma^z_i\sigma^z_{i+1}$$ has two degenerate ground states with a constant gap above (2 for periodic boundary conditions). This can be easily generalized to models with more than 2 states and correspondingly higher ground space degeneracy (Potts models).

• Thanks 'sure' and Norbert for answers. 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> - q|10>. Gap depends on parameter q and is gapped for q<1 . Reference: arxiv.org/abs/cond-mat/9512120 – anurag anshu Feb 12 '15 at 15:02