A simple mechanism for unstable degeneracies in local Hamiltonians

Abstract

If a local Hamiltonian eigenstate is mapped to another state by local operators commuting with the Hamiltonian terms, the latter is also an eigenstate. This basic observation implies a no-go result for both being a unique ground state and having a degeneracy protected against local perturbations.

1 page, 0 figures. Was this known or written somewhere? Comments are welcome. v2: Improved explanation and proof of instability corrected