Let's say I have a 2D quantum spin model of N spin-1/2 particles, with two terms:
$$ H = -J \sum_N \prod_{i \in G} \sigma^x_i - h \sum_N \prod_{i \in G'} \sigma^z_i $$
The first is a collection of products of $\sigma_x$ terms and the second of $\sigma_z$ terms. The two terms do not commute. I also impose PBC. The $G$ and $G'$ sets are arbitrary. Is it possible to engineer such a model with both phases having an exact ground state degeneracy even for finite system sizes (the same for both phases, or different if you want) for all values of the parameters of the two terms? How would I interpret these phases?
For example, this model does not have SSB in the infinite-size limit. It does not have topological order either, since one phase is classical and the other is (kind-of) a paramagnetic phase. It doesn't have SPT since I have an exact degeneracy for PBC... It does not have any gapless phase (most probably) either.
Since the two terms do not commute, there will be a quantum phase transition between the two phases of the model. What would be my approach for characterising the phases of the system? What can I say about the quantum phase transition? For example, I can use a Kramers-Wannier approach to show that the phase transition will be at $J = h$. However, my prerequisite is that there is everywhere a degeneracy, even at the critical point. How do I interpret then this point?
Are there known examples of quantum spin models with an exact degeneracy even for finite sizes with noncommuting terms? And what about quantum critical points with exact degeneracy?
