  We show that some gapped quantum many-body systems have a ground state
degeneracy that is stable to long-range (e.g., power-law) perturbations, in the
sense that any ground state energy splitting induced by such perturbations is
exponentially small in the system size. More specifically, we consider an Ising
symmetry-breaking Hamiltonian with several exactly degenerate ground states and
an energy gap, and we then perturb the system with Ising symmetric long-range
interactions. For these models we prove (1) the stability of the gap, and (2)
that the residual splitting of the low-energy states below the gap is
exponentially small in the system size. Our proof relies on a convergent
polymer expansion that is adapted to handle the long-range interactions in our
model. We also discuss applications of our result to several models of physical
interest, including the Kitaev p-wave wire model perturbed by power-law
density-density interactions with an exponent greater than 1.
